Cohomology rings of arrangements

Dissertationsschrift vorgelegt von Carsten Schultz

Fachbereich Mathematik und Informatik FU Berlin

Berlin, 2004

2 3 Betreuer und erster Gutachter: Prof. Dr. Elmar Vogt Freie Universität Berlin Fachbereich Mathematik und Informatik Arnimallee 14 D-14195 Berlin

vogt@math.fu-berlin.de

Zweiter Gutachter: Prof. Dr. Günter M. Ziegler Technische Universität Berlin Straße des 17. Juni 136 D-10623 Berlin

ziegler@math.tu-berlin.de

Tag der Disputation: 9. Februar 2005

Contents

Introduction  7
1 General Arrangements  11
1.1 Diagrams of spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.2 Diagrams of chain complexes . . . . . . . . . . . . . . . . . . . . . 15

Homology 15 Products 21

1.3 Arrangements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

Homotopy 22 Homology 25 Intersection products in manifolds 27

2 Linear and related arrangements  33
2.1 Homology and Homotopy . . . . . . . . . . . . . . . . . . . . . . . 33

ZŽ–maps 33 Spherical arrangements 35 Projective arrangements 38 Affine arrangements 42 Intersecting with a hyperplane 48

2.2 Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

Statement of results 52 Graded formulas 53 Inductive proofs of product formulas 55 Vanishing for affine arrangements 57

2.3 Products in projective arrangements . . . . . . . . . . . . . . . . . 58

An example of real projective arrangements 58 The product of two arrangements and general position 60 Recovering the direct sum decomposition 63 Vanishing for projective arrangements 65

2.4 Projective c-arrangements . . . . . . . . . . . . . . . . . . . . . . . 68

A presentation of the cohomology ring 68 The atomic complex 69 Proof of the presentation 72

Bibliography  77

6 CONTENTS

Introduction

This work is concerned with homotopy and homology properties of arrangements. An arrangement in a topological space X is a finite set A of subspaces of X. One goal in the study of arrangements is the description of the union ⋃ A and the complement X \ ⋃ A. For a linear subspace arrangement in real or projective space the intersection of the union of the arrangement with the unit sphere is also of interest. It is called the link of the arrangement. The link can be regarded as the union of an arrangement of spheres. By Alexander duality, the homology groups of the link determine the cohomology groups of the complement of the arrangement. They do not determine the multiplication in the cohomology ring.

Main result

This work treats the general theory of arrangements as well as properties of linear arrangements. Its main contribution however is the description of the cohomology ring of the complement of a complex projective arrangement. We will describe this result here in some detail, which will give us the opportunity to introduce some terminology.

Let V be a finite dimensional vector space over ℂ and let A be a finite set of linear subspaces of V . We denote the complex dimension of the projective space P V by n and set P A := {P A: A ∈ A}. We define Q := {⋂ M: M ⊂ A} and order this set by inclusion. The partially ordered set (poset) defined in this way is called the intersection poset of A. On it the dimension function d is defined by d(q) := dim P q. In particular d(⊤) = d(⋂ Ø) = d(V) = n. The result will be an explicit description of the cohomology ring of the complement of the projective arrangement P A in terms of the intersection poset and the dimension function.

Additively the cohomology of the complement is given by

H^{2n−i} (P V \ ⋃ P A) ≅ H_{i} (P V, ⋃ P A) ≅ ⨁_{k=0}^{n} H_{i−2k}(ΔQ_{[k,n]}, ΔQ_{[k,n)}). (1)

Here Q_{[k,n]} := {q ∈ Q: k ≤ d(q) ≤ n}, Q_{[k,n)} := {q ∈ Q: k ≤ d(q) < n} and for a poset P the order complex of P, i.e. the simplicial complex with simplices all chains in P, is denoted by ΔP. The first isomorphism above is Poincaré-Lefschetz-duality, and the second isomorphism will be described explicitly by maps h_{k}: H_{i−2k}(ΔQ_{[k,n]}, ΔQ_{[k,n)}) → H_{i} (P V, ⋃ P A).

On the ordered simplicial chain complex C_{∗} of ΔQ a product ˆ× is defined as the composition

ˆ×: C_{r}(ΔQ)⊗C_{s}(ΔQ) ⟶^{×} C_{r+s}(ΔQ×ΔQ) = C_{r+s}(Δ(Q×Q)) ⟶^{∧_{∗}} C_{r+s}(ΔQ), (2)

where ∧: Q×Q → Q is the map taking (u, v) to the minimum u∩v. The map ∧ is order preserving and hence a simplicial map. The product ˆ× induces products in homology. Denoting the intersection product on the homology of (P V, ⋃ P A) that corresponds via duality to the cup product on the cohomology of the complement P V \ ⋃ P A by • the cohomology ring will be fully described by (1) together with the following formula.

Theorem. For c ∈ H_{∗}(ΔQ_{[k,n]}, ΔQ_{[k,n)}) and d ∈ H_{∗}(ΔQ_{[l,n]}, ΔQ_{[l,n)})

h_{k}(c) • h_{l}(d) = \begin{cases}h_{k+l−n}(c ˆ× d), & k + l ≥ n, \\ 0, & k + l < n. \end{cases}  (3)

This will appear as Theorem 2.2.1 in this work. Since the intersection product is a composition of the cross product and the transfer of the diagonal map, the connection between it and the product ˆ× seems very natural in the light of (2).

History

We mention parts of the history of this subject that build a suitable context for the description of the content of this work.

Arnol’d has given a simple presentation in terms of generators and relations for the cohomology ring of the classifying space of the coloured braid group [Arn69]. This classifying space is the complement of the arrangement {{z: z_{i} = z_{j}} : i ≠ j} in ℂ^{n}, i.e. it is the complement of a linear complex hyperplane arrangement. This result has been extended to arbitrary linear complex hyperplane arrangements, where the cohomology is described by the Orlik-Solomon algebra of the intersection poset [OS80]. Several generalizations for other classes of complex linear subspace arrangements have been obtained afterwards.

For an arbitrary complex linear and projective subspace arrangement Goresky and MacPherson have given descriptions of the cohomology groups of the complement in terms of the intersection poset and the dimension function as an application of their stratified Morse theory [GM88]. A formula equivalent to (1) appears in that work. Ziegler and Živaljević have given a concrete homotopy equivalence between a space determined by the intersection poset and the dimension function and the link of a linear arrangement from which the homology formula can be read of [ZŽ93]. Their approach is to view an arrangement as a diagram of spaces (and inclusion maps). They have given an overview of homotopy theoretic tools which are useful in this setting and further applications to problems in combinatorics in [WZŽ99].

De Concini and Procesi produced rational models to show that the cohomology rings with rational coefficients of the complements of complex linear arrangements are determined by the intersection poset and the dimension function [DCP95]. Yuzvinsky used these models to endow the homology formulas given by Goresky and MacPherson for these arrangements with a combinatorially defined product which describes the cohomology ring of the complement of a complex linear arrangement [Yuz02]. This product is equivalent to the product ˆ× defined above under an isomorphism shifting dimensions by two, although the connection with the cross product was not made explicit. Starting from this description of the cohomology ring he was in a position to attack the problem of giving presentations in terms of generators and relations for special classes of arrangements in a purely combinatorial way and he generalized previous results [Yuz99]. His results were however, by the nature of the rational models which were at the foundations of this, confined to complex arrangements and cohomology with rational coefficients. The generalization of the product formula to integral coefficients and a class of real linear arrangements containing all complex linear arrangements was done independently by Deligne, Goresky and MacPherson [DGM00] and by de Longueville and the current author [dLS01]. The latter work uses quite explicit geometrical constructions for which it is important that the homology isomorphism are induced by the topological maps of Ziegler and Živaljević. It also introduces the product ˆ× in the form above.

Leitfaden

In Chapter 1 we deal with general arrangements in topological spaces. We first give a minimal overview over homotopy properties of diagrams of spaces. We then develop a corresponding theory of diagrams of chain complexes suitable to the study of homology properties of arrangements. This section features a spectral sequence that will be crucial in studying products later on. In the third section we show how to apply the results presented so far to the study of arrangements. Possibly new is the proof that a product formula like Yuzvinsky’s holds quite generally for the cohomology ring of the complement of an almost arbitrary arrangement in a manifold, albeit only in the graded object defined by the filtration of the cohomology ring induced by the spectral sequence. This graded formula will be be the basis for proofs of exact formulas in the second chapter. In the case of projective arrangements discussed above, such a graded formula would describe h_{k}(c) • h_{l}(d) only up to elements h_{i}(r_{i}) with i > k + l − n.

In Chapter 2 we enter the more concrete realm of linear arrangements. In Section 2.1 we prove several homology and also a few homotopy formulas for central linear, projective and affine arrangements, among them (1). While the isomorphisms and homotopy equivalences are constructed in a uniform manner. Still some redundancy is to be expected, as the aim is to demonstrate the use of the tools from Chapter 1 and connections between the different isomorphisms. There is probably nothing really new in that section, except perhaps that some things may not have been made quite so explicit before, including the connections between affine and projective arrangements and the homotopy equivalence for projective arrangements in Proposition 2.1.17. The topological maps forming this homotopy equivalance also induce the isomorphisms h_{k} defined above. To have such explicit descriptions of them makes our approach to the calculations of products possible.

In Section 2.2 we turn to determining the products in the cohomology rings of complements of linear arrangements. We first state the results for affine and projective arrangements. The product formula in Theorem 2.2.3 is the main result of [dLS01]. The corresponding formula for projective arrangements in Theorem 2.2.1 is the one discussed above. We then prove graded versions of these formulas by identifying them as special cases of a result from the first chapter. We show how the exact formulas can be derived from these by an inductive argument, if the vanishing of certain products can be guaranteed. An easy geometric argument then proves this vanishing for affine arrangements. For projective arrangements, this necessary vanishing is just the case k + l < n in (3) above. Its proof costs considerably more effort than in the affine case and takes up Section 2.3.

In Section 2.4 we derive from the product formula for projective arrangements thus proved a presentation of the cohomology ring of a projective c-arrangement. This is done by methods employed by Yuzvinsky in proving presentations for similar classes of linear arrangements.

Acknowledgements

My thanks go to Elmar Vogt for supervising this thesis and expressing his confidence in me during those inevitable phases of doubt.

From Mark de Longueville I learned about linear arrangements and the questions regarding their cohomology rings. Working together with him was a most enjoyable experience. When I later took up the study of arrangements again, he was always open to discussions. Thanks for your friendship, Mark!

With Stefan Geschke I had very pleasant mathematical conversations while running around a lake near the math department that helped keeping my appreciation of mathematics as a whole alive.

Chapter 1

General Arrangements

For a topological space X and a set A of subspaces of X (an arrangement in X) we will be interested in the homotopy type of ⋃ A and in the calculation of the homology groups of this space. We will also be interested in the space X \ ⋃ A, especially in its cohomology ring. If X is a manifold, we will describe the latter, using Poincaré duality, via the homology of the pair (X, ⋃ A) and intersection products therein.

The arrangement A defines a partially ordered set (poset), the intersection poset Q := {⋂ S: S ⊂ Q} ordered by inclusion. This is a special case of a diagram of spaces. We will more generally investigate diagrams over a small category ℭ (instead of the poset Q) where maps are not required to be inclusions. This additional generality will not complicate the proofs, and we hope that it will clarify the relevant concepts by allowing the reader the comparison with other special cases, for example that of the category ℭ being a group. We also envision applications of these tools to arrangements with group actions where ℭ would be the orbit category of the group, but this will not be explained in the current work.

1.1 Diagrams of spaces

In this section we will denote by 𝔗op a suitable category of topological spaces, such as the category presented in [Vog71].

Let ℭ be a (discrete) small category. We call an X ∈ 𝔗op^{ℭ}, i.e. a functor X: ℭ → 𝔗op, a ℭ-diagram of spaces. If the category ℭ is a group G, a G-diagram is a space with a G-operation defined on it. We will choose similar notation in such a way that an X ∈ 𝔗op^{ℭ} corresponds to a left operation and an X ∈ 𝔗op^{ℭ^{o}} corresponds to a right operation.

Diagrams of spaces have been used as a tool for studying homotopy types of arrangements in [ZŽ93] and [WZŽ99]. Since our main interest lies in computing homology groups and products, not in determining homotopy types, we summarize the needed results only briefly, giving proofs only where they illustrate concepts that will be useful later on, and using [HV92] and [FF89] as references. This is also hoped to motivate the material on diagrams of chain complexes in the next section, which proceeds analogously in some respects. In particular, we want to draw attention to the notions of free diagrams and ZŽ–maps.

The setting in which these results will be applied to arrangements is explained in Section 1.3.

1.1.1 Definition. Let X ∈ 𝔗op^{ℭ^{o}}, Y ∈ 𝔗op^{ℭ}. We define

X ×_{ℭ} Y := ∐_{q∈Obj ℭ}X(q) × Y (q)/ ∼,  (1.1)

where ∼ is the relation generated by (x, Y (f)y) ∼ (X(f)x, y) for x ∈ X(q), y ∈ Y (p), f ∈ ℭ(p, q).

1.1.2 Definition. For S ∈ 𝔗op^{Obj ℭ} we define i_{#}S ∈ 𝔗op^{ℭ} by

(i_{#}S)(q) := ∐_{p∈Obj ℭ} ∐_{f∈ℭ(p,q)}S(p)  (1.2)

for q ∈ Obj ℭ, and for f ∈ ℭ(q, q′) letting (i_{#}S)(f): (i_{#}S)(q) → (i_{#}S)(q′) map the copy of S(p) indexed by f identically to that indexed by gf. i_{#} : 𝔗op^{Obj ℭ} → 𝔗op^{ℭ} is made into a functor in the obvious way.

1.1.3 Definition. Let X ∈ 𝔗op^{ℭ}. We call X a free ℭ-diagram, if there exists a filtration X = ⋃_{n≥0} X_{n}, X_{n} ∈ 𝔗op^{ℭ}, such that X carries the final topology with respect to the inclusion maps X_{n} → X, X_{0}(q) = Ø for all q ∈ Obj ℭ, and for each n ≥ 1 exist A_{n}, B_{n} ∈ 𝔗op^{Obj ℭ}, a map j: B_{n} → A_{n} consisting of closed cofibrations, and a diagram

\xymatrix{i_{#}B_{n} \ar[d]_{i_{#}j} \ar[r] & X_{n−1} \ar[d]^{incl.} \\ i_{#}A_{n} \ar[r] & X_{n}}

which is a pushout diagram.

1.1.4 Remark. What we call a free ℭ-diagram is called a principal ℭ-space in [FF89]. Indeed it can be argued that the term free diagram would better be reserved to those in the image of the functor i_{#}. However, a chain complex is also called free, if it is free in every degree, and similarly free diagrams are made up of pieces in the image of i_{#}. Also compare with Definition 1.2.2.

1.1.5 Definition. We define a (ℭ × ℭ^{o})-diagram of simplicial sets by

\begin{matrix}Eℭ(q′, q)_{n} := \{(f_{0}, f_{1}, …, f_{n+1}): f_{i} ∈ ℭ, f_{0}f_{1} ⋯ f_{n+1} ∈ ℭ(q, q′)\} \\ Eℭ(g′, g)_{n} ((f_{0}, f_{1}, …, f_{n+1})) := (g′f_{0}, f_{1}, …, f_{n+1}g) \end{matrix}

with boundaries

d^{i}(f_{0}, f_{1}, …, f_{n+1}) := (f_{0}, …, f_{i}f_{i+1}, …, f_{n+1})

and degeneracies

s^{i}(f_{0}, f_{1}, …, f_{n+1}) := (f_{0}, …, f_{i}, id, f_{i+1}, …, f_{n+1}).

We define Eℭ ∈ 𝔗op^{ℭ×ℭ^{o}} to be the simplicial realization of this diagram.

1.1.6 Remark. We think of a X ∈ 𝔗op^{ℭ×ℭ^{o}} as being equipped with commuting right and left operations of ℭ. For example, for Y ∈ 𝔗op^{ℭ^{o}}, X ×_{ℭ} Y is defined and inherits the left operation, i.e. X ×_{ℭ} Y ∈ 𝔗op^{ℭ}.

1.1.7 Proposition. Let X ∈ 𝔗op^{ℭ}. Eℭ ×_{ℭ} X ∈ 𝔗op^{ℭ} is a free ℭ-diagram.

Sketch of proof. The idea is to choose the pairs (A_{n}(q), B_{n}(q)) of Definition 1.1.3 as (Δ^{n}, 𝔡Δ^{n}) × ∐_{q′∈Obj ℭ} ({(f_{1}, …, f_{n}): f_{1} ⋯ f_{n} ∈ ℭ(q′, q)} × X(q′)). □

1.1.8 Proposition. Let X ∈ 𝔗op^{ℭ}. A point of (Eℭ ×_{ℭ} X)(q) is determined by f_{0}, …, f_{n+1} with f_{0}f_{1} ⋯ f_{n+1} ∈ ℭ(q′, q), s ∈ Δ^{n} and x ∈ X(q′). Sending this point to X(f_{0}f_{1} ⋯ f_{n+1})(x) ∈ X(q) yields a well-defined map of ℭ-diagrams Eℭ ×_{ℭ} X → X.

1.1.9 Definition. For X ∈ 𝔗op^{ℭ} we define the colimit of X as (or identify it with) colim X := ∗×_{ℭ}X and we define the homotopy colimit of X as hcolim X := ∗ ×_{ℭ} Eℭ ×_{ℭ} X, where ∗ denoted the constant ℭ^{o}-diagram of spaces consisting of a single point.

The map from Proposition 1.1.8 induces a map hcolim X → colim X.

1.1.10 Remark. For X ∈ 𝔗op^{ℭ^{o}} we have hcolim X = ∗ ×_{ℭ^{o}} Eℭ^{o} ×_{ℭ^{o}} X ≈ X ×_{ℭ} Eℭ ×_{ℭ} ∗, and we will switch between both versions ad libitum.

1.1.11 Definition. Let X, Y ∈ 𝔗op^{ℭ}, f: X → Y a map of ℭ-diagrams. The map f is called a homotopy equivalence, if there exist g: Y → X, F: I × X → X, G: I × Y → Y , all of them maps of ℭ-diagrams, such that F is a homotopy from g ∘ f to id_{X} and G a homotopy from f ∘ g to id_{Y} .

1.1.12 Definition. Let X, Y ∈ 𝔗op^{ℭ}, f: X → Y a map of ℭ-diagrams. The map f is called a weak homotopy equivalence, if f(q): X(q) → Y (q) is a homotopy equivalence for all q ∈ Obj ℭ.

1.1.13 Proposition. The map Eℭ ×_{ℭ} X → X from Proposition 1.1.8 is a weak homotopy equivalence.

See Lemma 1.2.9 for a proof of an algebraic analogue.

The following Proposition is our main technical tool. A proof can be found in [FF89, Thm 4.3] where it is attributed to [BV73].

1.1.14 Proposition. Let X, Y ∈ 𝔗op^{ℭ}, f: X → Y a weak homotopy equivalence. If X and Y are free diagrams, then f is a homotopy equivalence.

1.1.15 Definition. Let X, Y ∈ 𝔗op^{ℭ^{o}}. We will call a weak equivalence f: X ×_{ℭ} Eℭ → Y a topological ZŽ–map.

1.1.16 Remark. A ZŽ–map f: X ×_{ℭ} Eℭ → Y yields homotopy equivalences X(q) ≃ (X ×_{ℭ} Eℭ)(q) ≃ Y (q) and should be seen as a collection of homotopy equivalences f_{q}: X(q) ⟶^{≃} Y (q) which not necessarily satisfy the equations f_{p} ∘ X(g) = Y (g) ∘ f_{q} that would make these maps into a map of ℭ-diagrams, but which satisfy these equations up to homotopies which fit together up to higher homotopies and so on. This will be made clearer for the algebraic analogue in Definition 1.2.12 and the calculations following it.

1.1.17 Proposition. Let X, Y ∈ 𝔗op^{ℭ^{o}} and f: X ×_{ℭ} Eℭ → Y be a ZŽ–map. If Y is a free ℭ^{o}-diagram, then the induced map hcolim X ≈ X ×_{ℭ} Eℭ ×_{ℭ} ∗ → Y ×_{ℭ} ∗ ≈ colim Y is a homotopy equivalence.

Proof. X ×_{ℭ}Eℭ is a free diagram by Proposition 1.1.7, Y by assumption. Therefore f is a homotopy equivalence by Proposition 1.1.14. It follows that the induced map is a homotopy equivalence. □

1.1.18 Remark. Our usage of the term ZŽ–map is motivated by the fact that the homotopy equivalence to the link of a linear arrangement from its combinatorially defined homotopy model given by Ziegler and Živaljević [ZŽ93] can be seen to arise in this way. We present their homotopy model in Proposition 2.1.10.

Because of Proposition 1.1.13 we can note a special case.

1.1.19 Corollary. Let X ∈ 𝔗op^{ℭ}. If X is a free ℭ-diagram, then the canonical map hcolim X → colim X is a homotopy equivalence. □

We conlude this short overview with a simple proposition that allows us to compose ZŽ–maps.

1.1.20 Proposition. Let f: X ×_{ℭ} Eℭ → Y be a ZŽ–map. The map

\begin{matrix}L(f): X ×_{ℭ} Eℭ & → Y ×_{ℭ} Eℭ \\ [(x, s)] & ↦ [(f(x, s), s)] \end{matrix}

is a well defined weak homotopy equivalence (and hence homotopy equivalence) of ℭ^{o}-diagrams which makes the diagram

\xymatrix{X ×_{ℭ} Eℭ \ar[r]^{L(f)} \ar[dr]_{f} & Y ×_{ℭ} Eℭ \ar[d] \\  & Y}  (1.3)

commute. Consequently, if g: Y ×_{ℭ} Eℭ → Z is another ZŽ–map, then so is g ∘ L(f): X ×_{ℭ} Eℭ → Z.

Proof. It is easily checked by computation that (1.3) is a well defined commutative diagram of ℭ^{o}-diagrams. Since f and the natural map Y ×_{ℭ} Eℭ → Y are weak homotopy equivalences, so is L(f). By Proposition 1.1.7 and Proposition 1.1.14 L(f) is even a homotopy equivalence. g ∘L(f) is a composition of weak homotopy equivalences and therefore a weak homotopy equivalence. □

1.2 Diagrams of chain complexes

We introduce notation and terminology for diagrams of chain complexes similar to those introduced in the previous section for diagrams of spaces. In particular, free diagrams and algebraic ZŽ–maps will be defined. These tools will be used later on to study homology groups of arrangements, starting in Section 1.3.

The main results of this section will be proved by use of the spectral sequences of a certain double complex. It is one of these spectral sequences that will also be the basis for the proof of the graded formula for intersection products in an arrangement in a manifold, Proposition 1.3.20.

Homology

Let ℭ be a small category, R a hereditary ring.

1.2.1 Definition. Let M ∈ R-𝔐od^{ℭ^{o}}, N ∈ R-𝔐od^{ℭ}. We define M ⊗_{ℭ} N ∈ R-𝔐od,

M ⊗_{ℭ} N := ⨁_{q∈Obj ℭ}M(q) ⊗ N(q)/K,

where K is the submodule generated by the elements a ⊗ N(f)(b) − M(f)(a) ⊗ b for a ∈ M(q), b ∈ N(p), f ∈ ℭ(p, q).

1.2.2 Definition. For S ∈ 𝔖et^{Obj ℭ}, we define F^{ℭ}S ∈ R-𝔐od^{ℭ} by

F^{ℭ}S(q) := F {(f, s): f ∈ ℭ(p, q), s ∈ S(p)} ,

and for g ∈ ℭ(q, q′) letting F^{ℭ}S(g) be the morphism sending (f, s) to (gf, s). M ∈ R-𝔐od^{ℭ} is called a free ℭ-diagram of abelian groups if M is isomorphic to F^{ℭ}S for some S, and X ∈ 𝔡R-𝔐od^{ℭ} is called a free ℭ-diagram of chain complexes, if X_{n} is a free diagram of abelian groups for every n ∈ ℤ.

1.2.3 Lemma. Let M ∈ R-𝔐od^{ℭ^{o}}, S ∈ 𝔖et^{Obj ℭ}. The map

⨁_{q∈Obj ℭ} ⨁_{s∈S(q)} M(q) → M ⊗_{ℭ} F^{ℭ}S

which sends an element m of the summand M(q) indexed by s ∈ S(q) to m ⊗ (id_{q}, s) is an isomorphism. □

1.2.4 Proposition. If N ∈ R-𝔐od^{ℭ} is a free ℭ–diagram, then the functor • ⊗_{ℭ} N: R-𝔐od^{ℭ^{o}} → R-𝔐od is exact.

Proof. This follows from Lemma 1.2.3. □

1.2.5 Definition. We define B(ℭ) ∈ 𝔡R-𝔐od^{ℭ×ℭ^{o}} by

B(ℭ)(q′, q)_{n} := F {q′ ⟵^{f_{0}} · ⟵^{f_{1}} · · · · · ⟵^{f_{n}} · ⟵^{f_{n+1}} q} ,

B(ℭ)(g′, g)(q′ ⟵^{f_{0}} · ⟵^{f_{1}} · · · · · ⟵^{f_{n}} · ⟵^{f_{n+1}} q) := p′ ⟵^{g′f_{0}} · ⟵^{f_{1}} · · · · · ⟵^{f_{n}} · ⟵^{f_{n+1}g} p,

for g ∈ ℭ(p, q), g′ ∈ ℭ(q′, p′),

𝔡(q′ ⟵^{f_{0}} · ⟵^{f_{1}} · · · · · ⟵^{f_{n}} · ⟵^{f_{n+1}} q) :=

∑_{k=0}^{n}(−1)^{k} q′ ⟵^{f_{0}} · · · · · ⟵^{f_{k−1}} · ⟵^{f_{k}f_{k+1}} · ⟵^{f_{k+2}} · · · · · ⟵^{f_{n+1}} q

1.2.6 Remark. B(ℭ)(q′, q) is the chain complex associated to the simplicial set Eℭ_{•} of Definition 1.1.5.

1.2.7 Lemma. B(ℭ) is a free ℭ × ℭ^{o}–diagram of chain complexes, B(ℭ)_{n} ≅

F^{ℭ×ℭ^{o}}S_{n} with S_{n}(q′, q) := {q′ ⟵^{f_{1}} ⋯ ⟵^{f_{n}} q}. □

1.2.8 Definition and Proposition. For K ∈ R-𝔐od^{ℭ} we define a map ε ∈ Hom_{ℭ}(B(ℭ) ⊗_{ℭ} K, K) by

ε: B(ℭ) ⊗_{ℭ} K → K
(⟵^{f_{0}} · · · · · ⟵^{f_{n+1}}) ⊗ k ↦ \begin{cases}0, & n > 0, \\ K(f_{0}f_{1})(k), & n = 0. \end{cases}

This is a chain map if the K in second position is regarded as a chain complex concentrated in degree zero.

Proof. Since

ε (𝔡 ((⟵^{f_{0}} · ⟵^{f_{1}} · ⟵^{f_{2}}) ⊗ k)) = ε ((⟵^{f_{0}f_{1}} · ⟵^{f_{2}}) ⊗ k − (⟵^{f_{0}} · ⟵^{f_{1}f_{2}}) ⊗ k)
= K(f_{0}f_{1}f_{2})(x) − K(f_{0}f_{1}f_{2})(x) = 0,

ε is a chain map. □

1.2.9 Lemma. The map ε from the preceding proposition is a chain homotopy equivalence. In other words, it is an acyclic resolution of K.

Proof. We define

L: (B(ℭ)_{r} ⊗_{ℭ} K)(q) → (B(ℭ)_{r+1} ⊗_{ℭ} K)(q)
(⟵^{f_{0}} · · · · · ⟵^{f_{r+1}}) ⊗ k ↦ (⟵^{id} · ⟵^{f_{0}} · · · · · ⟵^{f_{r+1}}) ⊗ k

and calculate for x = (⟵^{f_{0}} · · · · · ⟵^{f_{r+1}}) ⊗ k, that

(𝔡L + L𝔡)x = \begin{cases}x, & r > 0, \\ x − (⟵^{id} · ⟵^{id}) ⊗ ε(x), & r = 0, \end{cases}

proving that k ↦ (⟵^{id} · ⟵^{id}) ⊗ k is a homotopy inverse of ε. □

1.2.10 Proposition. Let X ∈ 𝔡R-𝔐od^{ℭ^{o}}, K ∈ R-𝔐od^{ℭ}, X_{p} = 0 for p < 0. If X is a free ℭ-diagram, then the map

H(X ⊗_{ℭ} B(ℭ) ⊗_{ℭ} K) ⟶^{H(id_{X}⊗ε)} H(X ⊗_{ℭ} K)

is an isomorphism.

Proof. Again viewing the right hand side K as a complex concentrated in degree zero, the map id_{X} ⊗ ε is a map of double complexes. Since X is free, the induced map ′′H(X ⊗_{ℭ}B(ℭ)⊗_{ℭ}K) → ′′H(X ⊗_{ℭ}K) = X ⊗_{ℭ}K, where ′′H denotes homology with respect to the second differential of a double complex, is isomorphic to id_{X} ⊗ H(ε): X ⊗_{ℭ} H(B(ℭ) ⊗_{ℭ} K) → X ⊗_{ℭ} H(K) = X ⊗_{ℭ} K by Proposition 1.2.4, and the latter map is an isomorphism by Lemma 1.2.9. By [God58, Thm I.4.3.1] it follows that H(id_{X} ⊗ ε) is also an isomorphism. □

1.2.11 Definition and Proposition. We define the diagonal chain map Δ ∈ Hom_{ℭ×ℭ^{o}}(B(ℭ), B(ℭ) ⊗_{ℭ} B(ℭ)) by

Δ(q′ ⟵^{f_{0}} p_{0} ⟵^{f_{1}} p_{1} ⋯ p_{n−1} ⟵^{f_{n}} p_{n} ⟵^{f_{n+1}} q) :=

∑_{k=0}^{n} q′ ⟵^{f_{0}} p_{0} ⋯ p_{k−1} ⟵^{f_{k}} p_{k} ⟵^{id_{p_{k}}} p_{k}⊗

p_{k} ⟵^{id_{p_{k}}} p_{k} ⟵^{f_{k+1}} p_{k+1} ⋯ p_{n} ⟵^{f_{n+1}} q.

Proof. This is a chain map by the usual calculation, additionally using

q′ ⟵^{f_{0}} p_{0} ⋯ p_{k−1} ⟵^{f_{k}} p_{k} ⊗ p_{k} ⟵^{id_{p_{k}}} p_{k} ⟵^{f_{k+1}} p_{k+1} ⋯ p_{n} ⟵^{f_{n+1}} q =

= q′ ⟵^{f_{0}} p_{0} ⋯ p_{k−2} ⟵^{f_{k−1}} p_{k−1} ⟵^{id_{p_{k−1}}} p_{k−1} ⊗ p_{k−1} ⟵^{f_{k}} p_{k} ⋯ p_{n} ⟵^{f_{n+1}} q

here. □

1.2.12 Definition and Proposition. Let X, Y ∈ 𝔡R-𝔐od^{ℭ^{o}}. A chain map between diagrams f ∈ Hom_{ℭ^{o}}(X ⊗_{ℭ} B(ℭ), Y) induces chain maps

f_{q}: X(q) → Y (q)
x ↦ f(x ⊗ q ⟵^{id_{q}} q ⟵^{id_{q}} q).

The maps H(f_{q}) form a homomorphism H(X) ⟶^{≅} H(Y), where H(X), H(Y) ∈ (R-𝔐od^{ℤ})^{ℭ^{o}}. We call the map f an algebraic ZŽ–map, if for every q ∈ ℭ and every R-module M the map H(f_{q}): H(X(q); M) → H(Y (q); M) is an isomorphism.

Proof. Using the isomorphism

η: Hom_{ℭ^{o}}(X ⊗_{ℭ} B(ℭ), Y) ⟶^{≅} Hom_{ℭ×ℭ^{o}}(B(ℭ), Hom(X, Y))

we can describe f_{q} by f_{q} = η(f) (q ⟵^{id_{q}} q ⟵^{id_{q}} q). Now

𝔡f_{q} = η(f) (𝔡 (q ⟵^{id_{q}} q ⟵^{id_{q}} q)) = 0,

i.e. f_{q} is a chain map. For k ∈ ℭ(p, q),

𝔡 (η(f) (⟵^{id_{q}} · ⟵^{k} · ⟵^{id_{p}})) = η(f) (⟵^{k} · ⟵^{id_{p}} − ⟵^{id_{q}} · ⟵^{k})
= f_{p} ∘ X(k) − Y (k) ∘ f_{q},

i.e. f_{p} ∘ X(k) ≃ Y (k) ∘ f_{q}. □

Looking at the preceding calculation from the other side, we get:

1.2.13 Proposition. Let X, Y ∈ 𝔡R-𝔐od^{ℭ^{o}}. Assume we are given chain maps f_{q}: X(q) → Y (q) for all q ∈ ℭ and for every k ∈ ℭ(p, q) a chain homotopy from Y (k) ∘ f_{q} to f_{p} ∘ X(k). If K_{kl} = Y (l) ∘ K_{k} + K_{l} ∘ X(k) whenever kl is defined, then the map

\begin{matrix}f: X ⊗_{ℭ} B(ℭ) & → Y \\ x ⊗ (⟵^{k_{1}} p_{0} ⟵^{k_{0}}) & ↦ f_{p_{0}}(x · k_{1}) · k_{0}, \\ x ⊗ (⟵^{k_{2}} p_{1} ⟵^{k_{1}} p_{0} ⟵^{k_{0}}) & ↦ (−1)^{|x|}K_{k_{1}}(x · k_{2}) · k_{0}, \\ x ⊗ (⟵^{k_{0}} ⋯ ⟵^{k_{n+1}}) & ↦ 0,  n > 1 \end{matrix}  (1.4)

is a chain map of ℭ^{o}–diagrams as in Definition 1.2.12.

Proof. f is a well-defined map of ℭ^{o}–diagrams, and to check that it is a chain map, it will suffice to show that η(f) is. We have

𝔡 (η(f) (⟵^{id_{q}} q ⟵^{id_{q}})) = 𝔡f_{q} = 0,
𝔡 (η(f) (⟵^{id_{q}} q ⟵^{k} p ⟵^{id_{p}})) = 𝔡K_{k} = k · k · f_{p} − f_{q} · k

= η(f) (⟵^{k} p ⟵^{id_{p}} − ⟵^{id_{q}} q ⟵^{k})

= η(f) (𝔡 (⟵^{id_{q}} q ⟵^{k} p ⟵^{id_{p}})) ,

η(f) (𝔡 (⟵^{id} · ⟵^{k} · ⟵^{l} · ⟵^{id})) = η(f) (⟵^{k} · ⟵^{l} · ⟵^{id}) − η(f) (⟵^{id} · ⟵^{kl} · ⟵^{id})

+ η(f) (⟵^{id} · ⟵^{k} · ⟵^{l})

= k · K_{l} − K_{kl} + K_{k} · l = 0,

proving this. □

We come to an algebraic analogue of Proposition 1.1.20.

1.2.14 Definition. Let X, Y ∈ 𝔡R-𝔐od^{ℭ^{o}} and f ∈ Hom_{ℭ^{o}}(X ⊗_{ℭ} B(ℭ), Y). We define L(f) ∈ Hom_{ℭ^{o}}(X ⊗_{ℭ} B(ℭ), Y ⊗_{ℭ} B(ℭ)) as the composition

L(f): X ⊗_{ℭ} B(ℭ) ⟶^{id⊗Δ} X ⊗_{ℭ} B(ℭ) ⊗_{ℭ} B(ℭ) ⟶^{f⊗id} Y ⊗_{ℭ} B(ℭ).

1.2.15 Proposition. Let X, Y, Z ∈ 𝔡R-𝔐od^{ℭ^{o}}, f ∈ Hom_{ℭ^{o}}(X ⊗_{ℭ} B(ℭ), Y), g ∈ Hom_{ℭ^{o}}(Y ⊗_{ℭ} B(ℭ), Z). Then h := g ∘ L(f) ∈ Hom_{ℭ^{o}}(X ⊗_{ℭ} B(ℭ), Z), L(h) = L(g) ∘ L(f), and if f and g are ZŽ–maps, then so is h.

Proof. L(h) = L(g) ∘ L(f) and h_{q} = g_{q} ∘ f_{q} are verified by calculation, using (id ⊗ Δ) ∘ Δ = (Δ ⊗ id) ∘ Δ for the first formula. □

1.2.16 Remark. Just in case that the equation L(g) ∘ L(f) = L(g ∘ L(f)) or the following diagram might seem oddly familiar to some readers, we remark that the functor T: X ↦ X ⊗_{ℭ} B(ℭ) together with the natural transformations ε′: T → 1 defined as in Definition 1.2.8 and id ⊗ Δ: T → T^{2} is a comonad.

We now obtain an algebraic analogue of Proposition 1.1.17.

1.2.17 Proposition. Let X, Y ∈ 𝔡R-𝔐od^{ℭ^{o}}, X_{r} = 0, Y_{r} = 0 for r < 0, K ∈ R-𝔐od^{ℭ}, f ∈ Hom_{ℭ^{o}}(X ⊗_{ℭ} B(ℭ), Y). Then the diagram

\xymatrix{X ⊗_{ℭ} B(ℭ) ⊗_{ℭ} K \ar[dr]^{f⊗id_{K}} \ar[r]^{L(f)⊗id_{K}} & Y ⊗_{ℭ} B(ℭ) ⊗_{ℭ} K \ar[d]^{id_{Y} ⊗ε} \\  & Y ⊗_{ℭ} K}

commutes. If f is a ZŽ–map, then H(L(f) ⊗ id_{K}) is an isomorphism. Therefore, H(f ⊗ id_{K}) is also an isomorphism, if additionally Y is a free diagram.

Proof. The commutativity of the diagram follows by calculation, the last sentence from Proposition 1.2.10. We have to check that H(L(f) ⊗ id_{K}) is an isomorphism for a ZŽ–map f.

Let x ∈ X(q)_{r}, c =⟵^{id_{q}} · ⟵^{f_{1}} ⋯ ⟵^{f_{n}} · ⟵^{id_{q}}∈ B(ℭ)(q, q′)_{s}, k ∈ K(q′). Then

(L(f) ⊗ id_{K})(x ⊗ c ⊗ k) ∈ ⨁_{i=0}^{s} Y_{p+i} ⊗_{ℭ} B(ℭ)_{s−i} ⊗_{ℭ} K. Since L(f) ⊗ id_{K} respects the filtration by s, it induces a homomorphism between the corresponding spectral sequences. The term of (L(f) ⊗ id_{K})(x ⊗ c ⊗ k) in X_{r} ⊗_{ℭ} B(ℭ)_{s} ⊗_{ℭ} K is f_{q}(x) ⊗ c⊗k. This describes the induced homomorphism between the E^{1}–terms ′H(X ⊗_{ℭ} B(ℭ)⊗_{ℭ}K) ≅ H(X ⊗K)⊗_{ℭ×ℭ^{o}} B(ℭ) and ′H(Y ⊗_{ℭ}B(ℭ)⊗_{ℭ}K) ≅ H(Y ⊗K)⊗_{ℭ×ℭ^{o}} B(ℭ) of the spectral sequences which is an isomorphism, because f is a ZŽ–map. By [God58, Thm I.4.3.1] it follows that H(L(f)⊗id_{K}) is also an isomorphism. □

Proposition 1.2.17 will later be used to substitute for a diagram Y a simpler diagram X. We now examine conditions on X which make this especially worthwhile.

1.2.18 Proposition. Let X ∈ 𝔡R-𝔐od^{ℭ^{o}}, K ∈ R-𝔐od^{ℭ}, K(q) a free R–module for all q ∈ ℭ. If X is isomorphic to a direct sum ⨁_{i∈I} X_{i}, X_{i} ∈ 𝔡R-𝔐od^{ℭ^{o}}, such that there exist n_{i} ∈ ℤ with H_{r}(X_{i}(q)) = 0 for all q ∈ ℭ, r ≠ n_{i}, then H(X ⊗_{ℭ} B(ℭ) ⊗_{ℭ} K) ≅ H(H(X) ⊗_{ℭ} B(ℭ) ⊗_{ℭ} K). This isomorphism is natural with respect to K and such that the diagram

[figure: H_{l}(X) ⊗_{ℭ} B(ℭ)_{0} ⊗_{ℭ} K; H_{l}(X ⊗_{ℭ} B(ℭ)_{0} ⊗_{ℭ} K); H_{0}(H_{l}(X) ⊗_{ℭ} B(ℭ) ⊗_{ℭ} K); ⨁_{r} H_{r}(H_{l−r}(X) ⊗_{ℭ} B(ℭ) ⊗_{ℭ} K); ≅; H_{l}(X ⊗_{ℭ} B(ℭ) ⊗_{ℭ} K)]  (1.5)

commutes.

1.2.19 Remark. Most of the time we will actually be able to choose an X with zero differentials, in which case the proposition becomes trivial.

Proof. Because of the additivity of all of the involved constructions, we may assume that there exists an n ∈ ℤ such that H_{r}(X) = 0 for all r ≠ n.

We will consider the spectral sequence of the double complex X ⊗_{ℭ} B(ℭ) ⊗_{ℭ} K associated to the filtration by the degree of B(ℭ). The E^{1}–term of this spectral sequence is ′H(X ⊗_{ℭ}B(ℭ) ⊗_{ℭ}K). Since B(ℭ) ⊗_{ℭ}K is a free ℭ–diagram, it follows that E_{r,s}^{1} ≅ H_{r}(X) ⊗_{ℭ} B(ℭ)_{s} ⊗_{ℭ} K and E_{r,s}^{2} ≅ H_{s}(H_{r}(X) ⊗_{ℭ} B(ℭ) ⊗_{ℭ} K). Since E_{r,s}^{2} = 0 for r ≠ n, E^{∞} = E^{2} and H_{n+s}(X ⊗_{ℭ} B(ℭ) ⊗_{ℭ} K) ≅ H_{s}(H_{n}(X) ⊗_{ℭ} B(ℭ) ⊗_{ℭ} K), proving the existence of the isomorphism.

The isomorphism is given by the maps

E_{n,s}^{∞} ≅ \frac{F_{s}H_{n+s}(X ⊗_{ℭ} B(ℭ) ⊗_{ℭ} K)}{F_{s−1}H_{n+s}(X ⊗_{ℭ} B(ℭ) ⊗_{ℭ} K)} ⟵^{≅} F_{s}H_{n+s}(X ⊗_{ℭ} B(ℭ) ⊗_{ℭ} K)

⟶^{≅} H_{n+s}(X ⊗_{ℭ} B(ℭ) ⊗_{ℭ} K),

where F denotes the induced filtration on the homology of the double complex. All of the maps are natural with respect to K.

The commutativity of the diagram is easily checked. □

1.2.20 Remark. If K does not consist of free modules, then the result will still hold, if we have that H_{r}(X_{i}(q); K(q′)) = 0 for all q, q′ ∈ ℭ, r ≠ n_{i} and if we replace H(H(X) ⊗_{ℭ} B(ℭ) ⊗_{ℭ} K) by H(H(X ⊗ K) ⊗_{ℭ×ℭ^{o}} B(ℭ)).

Products

When investigating intersection products in manifolds, many of the occurring chain complexes will carry products. We introduce some terminology to ease the description of these products.

We fix a functor λ: ℭ × ℭ → ℭ.

1.2.21 Definition. Let M ∈ R-𝔐od^{ℭ}. A λ-product on M is a natural transformation from the functor

ℭ × ℭ → R-𝔐od
(p, q) ↦ M(p) ⊗ M(q)

to the functor M ∘ λ.

1.2.22 Definition and Proposition. Let M ∈ R-𝔐od^{ℭ^{o}} and N ∈ R-𝔐od^{ℭ} be equipped with λ-products, written by juxtaposition. Then

(M ⊗_{ℭ} N) ⊗ (M ⊗_{ℭ} N) → (M ⊗_{ℭ} N)
(m ⊗ n) ⊗ (m′ ⊗ n′) ↦ mm′ ⊗ nn′

is well-defined, and we will usually equip M ⊗_{ℭ} N with this product. □

1.2.23 Definition and Proposition. We define the cross product × on B(ℭ), × ∈ Hom_{(ℭ×ℭ)×(ℭ×ℭ)^{o}}(B(ℭ) ⊗ B(ℭ), B(ℭ × ℭ)), by

(⟵^{f_{0}} · ⟵^{f_{1}} · · · · · ⟵^{f_{k}} · ⟵^{f_{k+1}}) ⊗ (⟵^{g_{0}} · ⟵^{g_{1}} · · · · · ⟵^{g_{l}} · ⟵^{g_{l+1}}) ↦

∑_{\begin{matrix}(i_{0},j_{0})<···<(i_{k+l},j_{k+l}) \\ (i_{0},j_{0})=(0,0) \\ (i_{k+l},j_{k+l})=(k,l) \\ i_{r+1}≤i_{r}+1 \\ j_{r+1}≤j_{r}+1 \end{matrix}} ε_{j_{0},...,j_{k+l}}^{i_{0},...,i_{k+l}} (⟵^{(f_{0},g_{0})} · ⟵^{h_{j_{0},j_{1}}^{i_{0},i_{1}}} · · · · · ⟵^{h_{j_{k+l−1},j_{k+l}}^{i_{k+l−1},i_{k+l}}} · ⟵^{(f_{k+1},g_{k+1})})

with h_{j,j+1}^{i,i} = (id, g_{j+1}), h_{j,j}^{i,i+1} = (f_{i+1}, id), ε_{0,1,...,l}^{0,0,...,0} = ε_{0,0,...,0}^{0,1,...,k} = 1, and the remaining ε_{j_{0},...,j_{k+l}}^{i_{0},...,i_{k+l}} ∈ {+1, −1} determined by the requirement that × be a chain map.

The functor λ induces a natural transformation λ_{∗} from B(ℭ×ℭ) to B(ℭ)∘(λ×λ^{o}) and the composition x ⊗ y ↦ λ_{∗}(x × y) is a λ-product (more precisely a (λ × λ^{o})-

product) on B(ℭ) that we will sometimes denote by ×^{λ}. □

1.3 Arrangements

Let X be a topological space and A a set of subspaces of X. We set Q := {⋂ S: S ⊂ A} and order Q by inclusion. The resulting partially ordered set Q will be considered a small category with a single arrow from p to q if p ≥ q. We define D ∈ 𝔗op^{Q^{o}} by D(p) := p and letting D(q ← p) be the inclusion from q to p.

1.3.1 Notation. The minimum map ∧: Q × Q → Q, p ∧ q = p ∩ q is order preserving, hence a functor. Q has a minimum ⋂ A and a maximum X = ⋂ Ø. These will be denoted by ⊥ and ⊤ respectively. For p, q ∈ Q, we denote by [p, q] the closed interval {x: p ≤ x ≤ q} and similar for open and half-open intervals. For a partially ordered set P, ΔP denotes the order complex of P, i.e. the simplicial complex with vertex set P and simplices all chains (totally ordered subsets) in P. By C(ΔP) we denote the ordered simplicial chain complex of the simplicial complex ΔP, for example B(Q)(p, q) = C_{∗}(Δ[p, q]).

Homotopy

When discussing homotopy properties of the arrangement A and the diagram D, we will assume that for all q ∈ Q the inclusion map ⋃_{p<q} D(p) → D(q) is a closed cofibration. The union on the left hand side can also be formulated as the colimit of the restrition of D to the poset {p: p < q}.

Under this hypothesis, we get the following proposition.

1.3.2 Proposition. Let Q be finite. Then D ∈ 𝔗op^{Q^{o}} is a free Q^{o}-diagram.

Proof. We enumerate Q as Q = {q_{1}, …, q_{m}} with i > j whenever q_{i} > q_{j} and define diagrams D^{k} ∈ 𝔗op^{Q^{o}} by

D^{k}(p) := ⋃ {D(q_{i}): i ≤ k, q_{i} ≤ p} = ⋃_{i≤k} D(p ∧ q_{i}).

These D^{k} form a filtration of D^{m} = D, D^{0}(p) = Ø for all p ∈ Q, and we will show that this filtration satisfies the conditions of Definition 1.1.3.

Let 0 < k ≤ m. For p ≥ q_{k}, we have D^{k}(p) = D^{k−1}(p) ∪ D(q_{k}) and

D^{k−1}(p) ∩ D(q_{k}) = ⋃_{i<k} D(p ∧ q_{i}) ∩ D(q_{k}) = ⋃_{i<k} D(q_{k} ∧ q_{i}) = ⋃_{q′<q_{k}} D(q′).

The equation in the middle is the one that is special to a diagram derived from an arrangement and would not hold for an arbitrary diagram of inclusion maps. The key point is that the right hand side is independent of p. It follows that

\xymatrix{⋃_{q′<q_{k}} D(q′) \ar[r] \ar[d] & D^{k−1}(p) \ar[d] \\ D(q_{k}) \ar[r] & D^{k}(p)}

is a pushout diagram. For p ≱ q_{k}, D^{k}(p) = D^{k−1}(p) and therefore

\xymatrix{Ø \ar[r] \ar[d] & D^{k−1}(p) \ar[d] \\ Ø \ar[r] & D^{k}(p)}

is a pushout diagram. Defining A_{k}, B_{k} ∈ 𝔗op^{Obj Q} by

A_{k}(p) := \begin{cases}D(q_{k}) & p = q_{k}, \\ Ø, & p ≠ q_{k}, \end{cases}  B_{k}(p) := \begin{cases}⋃_{q′<q_{k}} D(q′) & p = q_{k}, \\ Ø, & p ≠ q_{k}, \end{cases}

these combine to give a pushout diagram

\xymatrix{i_{#}B_{k} \ar[r] \ar[d] & D^{k−1} \ar[d] \\ i_{#}A_{k} \ar[r] & D^{k}}

of Q^{o}-diagrams. □

1.3.3 Remark. From the construction in the proof it follows immediately that a diagram obtained from D by restriction to a sub-poset {q_{1}, …, q_{m′}} with m′ < m and q_{i} as in the proof is also free.

1.3.4 Remark. If X allows a triangulation such that every A ∈ A is a subcomplex, a condition that we will assume later when considering intersection products but that is less natural when considering homotopy theory, then a proof more along the lines of Proposition 1.3.9 is available to show that the diagram D is free. This would involve defining A_{k}(p) to be the disjoint union of all k-simplices in D(p) not contained in any D(q) with q < p and B_{k}(p) the boundaries of those simplices. Finiteness of Q would not be needed.

Let us assume we are given the following data: A Q^{o}-diagram of spaces E ∈ 𝔗op^{Q^{o}} (in applications this will be easier to describe than D and possibly carry less information), and for all p ∈ Q maps

f^{p}: E(p) × Δ[p, ⊤] → X

with im f^{p}|_{E(p)×Δ[p,q]} ⊂ D(q) = q, f^{p}(·, ⟨p⟩): E(p) → D(p) = p a homotopy equivalence, and such that for q ≤ p the diagram

\xymatrix{ & E(p) × Δ[p, ⊤] \ar[dr]_{f^{p}} \\ E(p) × Δ[q, ⊤] \ar@{-->}[ur]_{id×incl} \ar[dr]^{E(p←q)×id} &  & X \\  & E(q) × Δ[q, ⊤] \ar[ur]^{f^{q}}}  (1.6)

commutes.

1.3.5 Proposition. In the above situation and with Q finite, the maps f^{p} induce a homotopy equivalence hcolim E′ ⟶^{≃} ⋃ A, where E′ is the diagram obtained by restricting E to Q \ {⊤}.

Proof. We set Q′ := Q \ {⊤} and denote the restriction of D to Q′ by D′. D is free by Proposition 1.3.2, and so is D′. The inclusion D′(q) → ⋃ A induce a map colim D′ → ⋃ A. Since ⋃ A = ⋃_{q∈Q′} D′(q) and all of the D′(q) are closed in ⋃ A, this map is a homeomorphism.

Since in Q viewed as a category the composition of two morphisms is never an identity unless one of the original morphisms was (indeed both of them), degenerate simplices may be omitted in the construction of EQ, and EQ(p, q) ≈ Δ[p, q]. Hence the maps f^{p} are exactly what it takes to define a map of Q^{o}-diagrams E ×_{Q} EQ → D, and they also define a map of Q^{o}′-diagrams E′ ×′_{Q} EQ′ → D′. The assumption that f^{p}(·, ⟨p⟩): E(p) → p is a homotopy equivalence means that this map is a ZŽ–map. The diagram D′ is free as shown in Proposition 1.3.2 respectively Remark 1.3.3. By Proposition 1.1.17 it follows that this ZŽ–map induces a homotopy equivalence hcolim E′ ⟶^{≃} colim D′ ≈ ⋃ A. □

1.3.6 Remark. For Proposition 1.3.5 it would not have been necessary to include ⋂ Ø = X in Q when defining the maps f^{p}. Indeed, it may seem like a nuissance that we have included ⋂ Ø = X in the definition of the intersection poset. However, when dealing with linear arrangements in Chapter 2 the constructed maps will naturally include the top element of the intersection poset and, more importantly, when turning to homology in the next section, we will also consider the relative case of (X, ⋃ A) and for this the top element will be needed.

1.3.7 Example. Let us assume that every intersection of elements of A is either empty or contractible. As discussed in the preceding remark, we allow us to ignore the empty intersection. We set

E(q) := \begin{cases}Ø, & D(q) = Ø \\ ∗, & D(q) ≠ Ø \end{cases}

and make E into a Q^{o}-diagram in the obvious and unique way. In this case hcolim E′ ≈ ΔN where N := {q ∈ Q: D(q) ≠ Ø, q < ⊤}. Maps f^{p} as above will automatically satisfy that f^{p}(·, ⟨p⟩) is a homotopy equivalence. Fulfilling the commutativity of (1.6) amounts to constructing a map h: ΔN → ⋃ A with h[⟨q_{0}, …, q_{r}⟩] ⊂ D(q_{r}) for all chains q_{0} < ⋯ < q_{r} with q_{0} ∈ N. Since D(q) is contractible for every q ∈ P, such a map is easily defined by recursion over the skeleton of ΔP. By Proposition 1.3.5 the map h is a homotopy equivalence.

1.3.8 Remark. The fact that the diagram D is free can be seen as the reason for the appearance of the intersection poset in descriptions of the homotopy type of ⋃ A or the homology formulas in the next section.

An alternative construction of a diagram D for which Proposition 1.3.5 holds is to define the poset Q to be the power set of A ordered by reverse inclusion and to set D(q) := ⋂ q. This is again a free diagram, which is proven by repeating the proof of Proposition 1.3.2 verbatim. Vassiliev calls the diagram defined via the power set of A the naive resolution and the diagram defined via the intersection poset the economical resolution [Vas01].

When using the naive resolution, the space ΔN in Example 1.3.7 becomes the nerve of A and the result of the example the Nerve Theorem.

We will only consider the economical resultion in the following. Some of the applications to linear arrangements would work equally well for both kinds of resolutions, while for some the economical resolution is more practical. We will look at this again in Remark 2.1.26.

We will meet the naive resolution briefly once again in Section 2.4 in the guise of the atomic complex.

Homology

We will be interested in describing H(⋃ A) and H(X, ⋃ A). For simplicity, we assume ⋃ A ≠ X. We assume the inclusion ∑_{A∈A} S(A) → S(⋃ A), where S denotes the singular chain complex, to induce an isomorphism in homology. Again we will write S(D) instead of S ∘ D for the diagram of chain complexes arising from D by applying the singular chain functor.

1.3.9 Proposition. S(D) ∈ 𝔡R-𝔐od^{Q^{o}} is a free Q^{o}–diagram.

Proof. Let p ∈ Q and σ: Δ^{r} → D(p) be a singular simplex. We set q_{σ} := ⋂ {q′ ∈ Q: im σ ⊂ q′}. Then q_{σ} ≤ p and σ ∈ S_{r}(q_{σ}). It follows that S_{r}(D) is freely generated by the system ({σ: Δ^{r} → D(p): q_{σ} = p})_{p∈Q}. □

1.3.10 Definition. We define K^{u}, K^{p} ∈ R-𝔐od^{Q} by

K^{u}(q) := \begin{cases}0, & q = ⊤, \\ R, & q < ⊤, \end{cases}  K^{u}(q′ → q) := \begin{cases}0, & q′ = ⊤, \\ id_{R}, & q′ < ⊤, \end{cases}

K^{p}(q) := \begin{cases}R, & q = ⊤, \\ 0, & q < ⊤, \end{cases}  K^{p}(q′ → q) := \begin{cases}id_{R}, & q′ = q = ⊤, \\ 0, & q < ⊤. \end{cases}

The notation is chosen because of the following connection of K^{u} and K^{p} with the singular chain complexes of the union ⋃ A and the pair (X, ⋃ A), respectively.

1.3.11 Proposition. The chain maps

S(D) ⊗_{Q} K^{u} → S (⋃ A) ,  S(D) ⊗_{Q} K^{p} → S (X, ⋃ A)
c ⊗ k ↦ kc  c ⊗ k ↦ kc

are well defined and they induce isomorphisms H(S(D) ⊗_{Q} K^{u}) ≅ H(⋃ A) and H(S(D) ⊗_{Q} K^{p}) ≅ H(X, ⋃ A).

Proof. By the proof of Proposition 1.3.9 and Lemma 1.2.3,

S_{r}(D) ⊗_{Q} K^{u} ≅ ⨁_{p<⊤} ⨁_{\begin{matrix}σ∈S_{r}(X) \\ q_{σ}=p \end{matrix}} R = ⨁_{\begin{matrix}σ∈S_{r}(X) \\ q_{σ}∈Q\\{⊤\} \end{matrix}} R = ∑_{A∈A} S(A) =: SA,

and the first induced map factorizes as H(S_{r}(D) ⊗_{Q} K^{u}) ⟶^{≅} H(SA) ⟶^{≅} H(⋃ A).

Similarly, S_{r}(D) ⊗_{Q} K^{p} is free on the r–simplices σ with q_{σ} = ⊤, and the second induced map factorizes as H(S_{r}(D) ⊗_{Q} K^{p}) ⟶^{≅} H(SX/SA) ⟶^{≅} H(X, ⋃ A). □

1.3.12 Proposition. In the situation described in (1.6), with the condition on f^{p}(·, ⟨p⟩): E(p) → p weakened to induce an isomorphism in homology,

g: S(E) ⊗_{Q} B(Q) → S(D)
c ⊗ p ← q_{0} ← ⋯ ← q_{n} ← p′ ↦ f_{∗}^{p}(c × ⟨q_{0}, …, q_{n}⟩),

defines a ZŽ–map. For K ∈ R-𝔐od^{Q} the map g_{∗}: H(S(E) ⊗_{Q} B(Q) ⊗_{Q} K) → H(S(D) ⊗_{Q} K) is an isomorphism.

Proof. Because im f^{p}|_{E(p)×Δ[p,p′]} ⊂ p′, f_{∗}^{p}(c × ⟨q_{0}, …, q_{n}⟩) is in S(D)(p) = S(p). It is well-defined because of the commutativity of (1.6). The map is a map of Q^{o}–diagrams, because the right hand side is independent of p′. That it is a chain map is now easily checked.

The map S(E)(p) → S(D)(p) = S(p), c ↦ g(c ⊗ p ← p ← p) = f_{∗}^{p}(c × ⟨p⟩) induces an isomorphism in homology by assumption, so g is a ZŽ–map. The map g_{∗} is an isomorphism by Proposition 1.2.17 and Proposition 1.3.9. □

In this situation, one may be lucky and able to prove that H(S(E)⊗_{Q}B(Q)⊗_{Q}K) is isomorphic to H(H(E) ⊗_{Q} B(Q) ⊗_{Q} K), e.g. by Proposition 1.2.18. It then follows that H(S(D) ⊗_{Q} K) ≅ H(H(D) ⊗_{Q} B(Q) ⊗_{Q} K).

The preceding results, and those in the section to come, are easily extended to the relative case. We will formulate and prove only the key step.

1.3.13 Proposition. Let Y ⊂ X and assume that the inclusion maps

S(Y) + S (⋃ A) → S (Y ∪ ⋃ A) ,
∑_{A∈A} S(A ∩ Y) → S (Y ∩ ⋃ A)

also induce isomorphisms in homology. Let D′ be the Q^{o}–diagram of pairs of spaces defined by D′(q) := (q, q ∩ Y). Then S(D′) is a free Q^{o}–diagram, and the chain maps

S(D′) ⊗_{Q} K^{u} → S (⋃ A, Y ∩ ⋃ A) , S(D′) ⊗_{Q} K^{p} → S (X, Y ∪ ⋃ A)
c ⊗ k ↦ kc  c ⊗ k ↦ kc

are well defined and induce isomorphisms H (⋃ A, Y ∩ ⋃ A) ≅ H(S(D′) ⊗_{Q} K^{u}) and H (X, Y ∪ ⋃ A) ≅ H(S(D′) ⊗_{Q} K^{p}).

Let Q̄ := {p ∈ Q: D(p) ⊄ Y } and D̄′ the Q̄^{o}–diagram obtained by restricting D′. Then S(D̄′) is a free diagram and for any K ∈ 𝔄b^{Q} the obvious map S(D̄′) ⊗Q̄ K̄ → S(D′) ⊗_{Q} K, where K̄ is the restriction of K, is an isomorphism. In particular, H (⋃ A, Y ∩ ⋃ A) ≅ H(S(D̄′) ⊗Q̄ K^{u}) and H (X, Y ∪ ⋃ A) ≅ H(S(D̄′) ⊗Q̄ K^{p}).

Proof. Taking up the notation of the proof of Proposition 1.3.9, S_{r}(D′) is free on the system ({σ: Δ^{r} → D(p): q_{σ} = p, im σ ⊄ Y })_{p∈Q}, and as in Proposition 1.3.11

the first induced map factorizes as H(S(D′) ⊗_{Q} K^{u}) ⟶^{≅} H(SA/(SA ∩ SY)) ⟶^{≅} H (⋃ A, Y ∩ ⋃ A). S(D′)⊗_{Q} K^{p} is free on the singular simplices neither in Y nor in any of the A ∈ A, and the second induced map factorizes as H(S(D′)⊗_{Q}K^{p}) ⟶^{≅}

H(SX/(SY ∪ SA)) ⟶^{≅} H (X, Y ∪ ⋃ A).

To justify the claims regarding Q̄, it suffices to remark that the free diagram S_{r}(D′) has no generators for p ∈ Q \ Q̄. □

Intersection products in manifolds

If X is a compact n-dimensional manifold oriented over R, we are interested in the intersection products • defined by commutativity of

\xymatrix{H_{k}(X, ⋃ A) ⊗ H_{l}(X, ⋃ A) \ar[r]^{•} & H_{k+l−n}(X, ⋃ A) \\ H^{n−k}(X \ ⋃ A) ⊗ H^{n−l}(X \ ⋃ A) \ar[r]^{⌣} \ar[u]^{⌢[X]}_{≅} & H^{2n−k−l}(X \ ⋃ A) \ar[u]^{⌢[X]}_{≅}}

and

\xymatrix{H_{k}(⋃ A) ⊗ H_{l}(⋃ A) \ar[r]^{•} & H_{k+l−n}(⋃ A) \\ H^{n−k}(X, X \ ⋃ A) ⊗ H^{n−l}(X, X \ ⋃ A) \ar[r]^{⌣} \ar[u]^{⌢[X]}_{≅} & H^{2n−k−l}(X, X \ ⋃ A). \ar[u]^{⌢[X]}_{≅}}

For Poincaré duality to hold and for technical reasons, we assume X to allow a triangulation such that all A ∈ A are subcomplexes.

In this section we will see what information about the intersection products can obtained algebraically without special geometric knowledge of the class of arrangements at hand. For linear arrangements, this will yield the graded formulas of Section 2.2.

For the description of these products it will be important that there is a product on C_{∗}(ΔQ).

1.3.14 Definition and Remark. Let c ∈ C_{r}(ΔQ), d ∈ C_{s}(ΔQ). Then we have c × d ∈ C_{r+s}(ΔQ × ΔQ) = C_{r+s}(Δ(Q × Q)) and ∧_{∗}(c × d) ∈ C_{r+s}(Q), since ∧: Q × Q → Q is order preserving and hence a simplicial map Δ(Q × Q) → ΔQ. If c = ⟨p_{0}, …, p_{r}⟩ and d = ⟨q_{0}, …, q_{s}⟩, then ∧_{∗}(c × d) is a linear combination of simplices with first vertex p_{0} ∧ q_{0} and last vertex p_{r} ∧ q_{s}. This specializes Definition 1.2.23 with ∧ for λ, and as there we will set c ˆ× d := ∧_{∗}(c × d). The multiplication in R defines ∧-products (see Definition 1.2.21) on K^{u} and K^{p} in the obvious way. These products and Definition 1.2.22 will be used to define products on several chain complexes.

1.3.15 Proposition. Let K be equipped with a ∧-product in the above situation. The spectral sequence of the filtration of S(D) ⊗_{Q} B(Q) ⊗_{Q} K by the grading of B(Q) can be made into a multiplicative E^{1}-spectral sequence with the multiplication on E^{1} isomorphic to the multiplication on H(D) ⊗_{Q} B(Q) ⊗_{Q} K given by

(a ⊗ ⟨p_{0}, …, p_{r}⟩ ⊗ m) ⊗ (b ⊗ ⟨q_{0}, …, q_{s}⟩ ⊗ m′)

↦ (−1)^{r(n−l)}[(a • b) ⊗ (⟨p_{0}, …, p_{r}⟩ ˆ×⟨q_{0}, …, q_{s}⟩) ⊗ (m · m′), (1.7)

where a ∈ H_{k}(D(p)), b ∈ H_{l}(D(q)), a • b ∈ H_{k+l−n}(D(p ∧ q)), m ∈ K(p′), m′ ∈ K(q′), m · m′ ∈ K(p′ ∧ q′). The multiplication is a chain map of degree (−n, 0). If K = K^{u} or K = K^{p} the multiplication on E^{∞} is induced, via the isomorphism H(S(D) ⊗_{Q} B(Q) ⊗_{Q} K^{u}) ≅ H(⋃ A) or H(S(D) ⊗_{Q} B(Q) ⊗_{Q} K^{p}) ≅ H(X, ⋃ A) respectively, by the intersection product on H(⋃ A) or H(X, ⋃ A).

1.3.16 Remark. The intersection product a•b in the above proposition is defined by commutativity of

\xymatrix{H_{k}(A) ⊗ H_{l}(B) \ar[r]^{•} & H_{k+l−n}(A ∩ B) \\ H^{n−k}(X, X \ A) ⊗ H^{n−l}(X, X \ A) \ar[r]^{⌣} \ar[u]^{⌢[X]}_{≅} & H^{2n−k−l}(X, (X \ A) ∪ (X \ B)). \ar[u]^{⌢[X]}_{≅}}

Proof of Proposition 1.3.15

We will from now on consider X to be triangulated by a barycentric subdivision of a triangulation of which all A ∈ A are subcomplexes. This will make all p ∈ Q full subcomplexes of X. We will denote the face poset of this triangulation by F X and by C(F X) the chain complex of ascending (from 0-simplices to n-simplices) chains in F X.

1.3.17 Definition and Proposition. For a subcomplex A of X, cap products

C^{r}(F X, F X \ F A) ⊗ C_{s}(F X) ⟶^{⌢} C_{s−r}(F A)
C^{r}(F X \ F A) ⊗ C_{s}(F X) ⟶^{⌢} C_{s−r}(F X, F A)
h ⊗ ⟨f_{0}, …, f_{s}⟩ ↦ (−1)^{r(s−r)}h(⟨f_{s−r}, …, f_{s}⟩)⟨f_{0}, …, f_{s−r}⟩

are defined, where

C^{r}(F X, F X \ F A) = ker (Hom(C_{r}(F X), R) → Hom(C_{r}(F X \ F A), R)) ,
C_{s−r}(F X, F A) = coker (C_{s−r}(F A) → C_{s−r}(F X)) .

Proof. If ⟨f_{0}, …, f_{s−r}⟩ is not in C_{∗}(F A), then f_{s−r} is not in A and therefore ⟨f_{s−r}, …, f_{s}⟩ is in C_{∗}(F X \ F A), so that h(⟨f_{s−r}, …, f_{s}⟩) = 0 for the first kind of product or h(⟨f_{s−r}, …, f_{s}⟩) is defined for the second kind of product. □

Since Δ(F X) is just the barycentric subdivison of X, we have H(C(F X)) ≅ H(X). Let o ∈ C_{n}(F X) represent the orientation class [X] ∈ H_{n}(X). Regarding C(F A) as a subcomplex of the singular chain complex S(A), this yields a map C(F X, F X \ F A) ⟶^{⌢o} S(A) which induces an isomorphism in homology, if A is a full subcomplex. Δ(F X \ F A) is the subcomplex of the barycentric subdivision of X that consists of all simplices which do not meet A. It is the complement of an open normal neighbourhood of A.

1.3.18 Definition and Proposition. If A, B are subcomplexes of X, a cup product

C^{r}(F X, F X \ F A) ⊗ C^{s}(F X, F X \ F B) → C^{r+s}(F X, F X \ F (A ∩ B))
g ⊗ h ↦ g ⌣ h,

(g ⌣ h)(⟨f_{0}, …, f_{r+s}⟩) := (−1)^{rs}g(⟨f_{0}, …, f_{r}⟩)h(⟨f_{r}, …, f_{r+s}⟩), is defined.

Proof. If ⟨f_{0}, …, f_{r+s}⟩ is in C_{∗}(F X \ F (A ∩ B)), then f_{0} ∉ F (A ∩ B) and therefore either f_{0} ∉ F A and ⟨f_{0}, …, f_{r}⟩ ∈ C(F X \ F A) or f_{0} ∉ F B and ⟨f_{r}, …, f_{r+s}⟩ ∈ C(F X \ F B). In either case (g ⌣ h)(⟨f_{0}, …, f_{r+s}⟩) = 0. □

We now define Y ∈ 𝔡R-𝔐od^{Q^{o}} by Y (p)_{r} := C^{−r}(F X, F X \ F p). We equip this with the ∧–product given by the cup product just defined. This also defines products on Y ⊗_{Q} K and Y ⊗_{Q} B(Q) ⊗_{Q} K by Definition 1.2.22 and Definition 1.2.23.

The product on the double complex Y ⊗_{Q} B(Q) ⊗_{Q} K makes its second spectral sequence into a multiplicative spectral sequence with the multiplication on the E^{1}-term H(Y) ⊗_{Q} B(Q) ⊗_{Q} K isomorphic to

[α ⊗ ⟨p_{0}, …, p_{r}⟩ ⊗ m] ⊗ [β ⊗ ⟨q_{0}, …, q_{s}⟩ ⊗ m′]

↦ (−1)^{rl}[(α ⌣ β) ⊗ (⟨p_{0}, …, p_{r}⟩ ˆ×⟨q_{0}, …, q_{s}⟩) ⊗ (m · m′),

where α ∈ H^{k}(X, X \ p), β ∈ H^{l}(X, X \ q), α ⌣ β ∈ H^{k+l}(X, X \ (p ∩ q)), m ∈ K(p′), m′ ∈ K(q′), m · m′ ∈ K(p′ ∧ q′).

Now ⌢ o is a Q^{o}-chain-map (of degree n) from Y to S(D), inducing isomorphisms

H^{∗}(X, X \ p) ≅ H(Y (p)) ⟶_{≅}^{⌢[X]} H(S(D(p))) = H_{∗}(p)

for all p. It therefore induces an isomorphism between the second spectral sequences of the double complexes Y ⊗_{Q} B(Q) ⊗_{Q} K and S(D) ⊗_{Q} B(Q) ⊗_{Q} K from the E^{1}-terms on. We use this isomorphism to make the spectral sequence of S(D) ⊗_{Q} B(Q) ⊗_{Q} K multiplicative. Since ⌢ [X] takes cup products into intersection products, this already proves the first part of the proposition.

1.3.19 Proposition. The maps

Y ⊗_{Q} K^{u} → C^{∗}(F X, F X \ F ⋃ A),
Y ⊗_{Q} K^{p} → C^{∗}(F X \ F ⋃ A),
[f ⊗ k] ↦ kf

are well defined and respect products.

Proof. Y (⊤) ⊗ K^{u}(⊤) = 0 and for q < ⊤ the complex Y (q) = C^{∗}(F X, F X \ F q) is a subcomplex of C^{∗}(F X, F X \ F ⋃ A), since q ⊂ ⋃ A. Therefore the map Y ⊗_{Q} K^{u} → C^{∗}(F X, F X \ F ⋃ A) is well defined.

Y (q) ⊗ K^{p}(q) = 0 for q < ⊤ and Y (⊤) = C^{∗}(F X) restricts to C^{∗}(F X \ F ⋃ A). Let q < ⊤, k ∈ K^{p}(⊤), f ∈ Y (q) = C^{∗}(F X, F X \ F q). Then Y (q ← ⊤)f ⊗ k = f ⊗ K^{p}(q ← ⊤)k = 0 and f restricts to 0 in C^{∗}(F X \ F ⋃ A). Therefore the map Y ⊗_{Q} K^{p} → C^{∗}(F X \ F ⋃ A) is well defined.

Both maps respect products because of the naturality of cup products. □

The multiplication on E^{∞} is induced by the multiplication on the homology of the double complex. In the commutative diagram

\xymatrix{H(Y ⊗_{Q} B(Q) ⊗_{Q} K^{p}) \ar[r]^{H((⌢o)⊗id⊗id)}_{≅} \ar[d]^{H(id_{Y} ⊗ε)} & H(S(D) ⊗_{Q} B(Q) ⊗_{Q} K^{p}) \ar[d]^{H(id_{S(D)}⊗ε)}_{≅} \\ H(Y ⊗_{Q} K^{p}) \ar[r]^{H((⌢o)⊗id)}_{≅} \ar[d] & H(S(D) ⊗_{Q} K^{p}) \ar[d]_{≅} \\ H^{∗}(X \ ⋃ A) \ar[r]^{⌢[X]} & H_{∗}(X, ⋃ A)}

the maps on the left are ring homomorphisms, while the map at the bottom takes cup products into cap products. This proves the second part of the proposition for K^{p} and H_{∗}(X, ⋃ A). The corresponding diagram with K^{u}, H^{∗}(X, X \ ⋃ A), and H_{∗}(⋃ A) completes the proof of Proposition 1.3.15 □

Product formulas

We apply Proposition 1.3.15 to a class of arrangements for which the E^{2}-term of the spectral sequence is isomorphic to the homology of the arrangement.

1.3.20 Proposition. Assume that there is Z ∈ 𝔡R-𝔐od^{Q^{o}} satisfying the condition of Proposition 1.2.18 and a ZŽ–map ζ: Z ⊗_{Q} B(Q) → S(D). Then

\begin{aligned}H(H(D) ⊗_{Q} B(Q) ⊗_{Q} K) & ⟵^{≅} H(H(Z) ⊗_{Q} B(Q) ⊗_{Q} K) \\ & ⟶_{α}^{≅} H(Z ⊗_{Q} B(Q) ⊗_{Q} K) \\ & ⟶^{≅} H(S(D) ⊗_{Q} B(Q) ⊗_{Q} K), \end{aligned}  (1.8)

the isomorphism α being that from Proposition 1.2.18 (and trivial, if the boundary map in Z equals zero) and the other two induced by ζ: The first arrow by the isomorphism H(Z) ≅ H(D) described in Definition 1.2.12 and the last one by Proposition 1.2.17. We denote the composition of these isomorphisms by ϕ̃ and decompose the resulting isomorphism ϕ: H(H(D)⊗_{Q} ⊗B(Q)⊗_{Q} K^{p}) → H_{∗}(X, ⋃ A) as ϕ = ∑_{k} ϕ_{k} with

ϕ_{k}: H_{r}(H_{k}(D) ⊗_{Q} B(Q) ⊗_{Q} K^{p}) → H_{k+r} (X, ⋃ A) .  (1.9)

Then

ϕ_{k}(x) • ϕ_{l}(y) − ϕ_{k+l−n}(x · y) ∈ ⨁_{i>k+l−n} im ϕ_{i},  (1.10)

where x · y denotes the product induced by (1.7).

Proof. The filtration defining the spectral sequence in Proposition 1.3.15 induces a filtration on H(S(D) ⊗_{Q} B(Q) ⊗_{Q} K^{p}) and we will first identify its image in H_{∗} (X, ⋃ A). On H(Z ⊗_{Q} B(Q)⊗_{Q} K^{p}) the filtration given by the degree of B(Q) is

F_{k}(H_{t}(Z ⊗_{Q} B(Q) ⊗_{Q} K^{p})) = ⨁_{i=0}^{k} α [H_{i}(H_{t−i}(Z) ⊗_{Q} B(Q) ⊗_{Q} K^{p})] .

The map L(ζ) ⊗ id_{K^{p}} induces an isomorphism respecting filtrations, therefore the filtration on H_{∗} (X, ⋃ A) induced by that on H(S(D) ⊗_{Q} B(Q) ⊗_{Q} K^{p}) is

F_{k} (H_{t} (X, ⋃ A)) = ⨁_{i=0}^{k} ϕ_{t−i} [H_{i}(H_{t−i}(D) ⊗_{Q} B(Q) ⊗_{Q} K^{p})] .

For x ∈ H_{r}(H_{k}(D)⊗_{Q}B(Q)⊗_{Q}K^{p}), ϕ̃(x) ∈ F_{r}(H_{k+r}(S(D)⊗_{Q}B(Q)⊗_{Q}K^{p})). The E^{∞}-term F_{r}(H_{k+r}(S(D) ⊗_{Q} B(Q) ⊗_{Q} K^{p}))/F_{r−1}(H_{k+r}(S(D) ⊗_{Q} B(Q) ⊗_{Q} K^{p})) equals the E^{2}-term H_{r}(H_{k}(D)⊗_{Q}B(Q)⊗_{Q}K^{p}), and the class that ϕ̃(x) represents in the E^{∞}-term is again x. From this and Proposition 1.3.15 it follows that for y ∈ H_{s}(H_{l}(D) ⊗_{Q} B(Q) ⊗_{Q} K^{p}) we have

ϕ_{k}(x) • ϕ_{l}(y) − ϕ_{k+l−n}(x · y) ∈ F_{r+s−1} (H_{k+l−n+r+s} (X, ⋃ A))

⊂ ⨁_{i=0}^{r+s−1} im ϕ_{k+l−n+r+s−i} = ⨁_{i=1}^{r+s} im ϕ_{k+l−n+i} ⊂ ⨁_{i>k+l−n} im ϕ_{i}

as stated. □

In Proposition 1.3.15 the case of K = K^{u}, i.e. of intersection products in H_{∗}(⋃ A), is the less interesting one.

1.3.21 Proposition. The product

H_{r}(H_{k}(D) ⊗_{Q} B(Q) ⊗_{Q} K^{u}) ⊗ H_{s}(H_{l}(D) ⊗_{Q} B(Q) ⊗_{Q} K^{u}) → H_{r+s}(H_{k+l−n}(D) ⊗_{Q} B(Q) ⊗_{Q} K^{u})

is zero except for r = s = 0.

Proof. Let x ∈ H_{r}(H_{k}(D) ⊗_{Q} B(Q) ⊗_{Q} K^{u}), y ∈ H_{s}(H_{l}(D) ⊗_{Q} B(Q) ⊗_{Q} K^{u}). The sequence

H_{r+1}(H_{k}(D) ⊗_{Q} B(Q) ⊗_{Q} R^{p}) ⟶^{𝔡} H_{r}(H_{k}(D) ⊗_{Q} B(Q) ⊗_{Q} K^{u})

→ H_{r}(H_{k}(D) ⊗_{Q} B(Q) ⊗_{Q} K)

is exact. If r > 0, then H_{r}(H_{k}(D) ⊗_{Q} B(Q) ⊗_{Q} K) = 0 and therefore z ∈ H_{r+1}(H_{k}(D) ⊗_{Q} B(Q) ⊗_{Q} R^{p}) exists with 𝔡z = x. It is easily checked that with the obvious definition of the ∧–product R^{p} ⊗ R^{u} → R^{u} it follows that xy = (𝔡z)y = 𝔡(zy) = 0. □

We will, however, often be able to get a better result on intersection products in H_{∗}(⋃ A) without resorting to Proposition 1.3.15.

1.3.22 Proposition. We assume the situation of Proposition 1.3.20 and define ϕ_{k}: H_{r}(H_{k}(D) ⊗_{Q} B(Q) ⊗ K^{u}) → H_{k+r}(⋃ A) analogously. Then we have ϕ_{k}(x) • ϕ_{l}(y) = 0, unless |x| = |y| = 0, in which case the product is determined by

ϕ_{k}([a ⊗ ⟨p⟩ ⊗ m]) • ϕ_{l}([b ⊗ ⟨q⟩ ⊗ m′]) = ϕ_{k+l−n}([(a • b) ⊗ ⟨p ∧ q⟩ ⊗ mm′]).

Proof. Assume that |x| > 0. As in the preceding proof, x maps to zero in H(H(D)⊗_{Q}B(Q)⊗_{Q}R). Since the isomorphisms in (1.8) are natural with respect to K, the diagram

\xymatrix{H(H(D) ⊗_{Q} B(Q) ⊗_{Q} R^{u}) \ar[r] \ar[d]^{ϕ} & H(H(D) ⊗_{Q} B(Q) ⊗_{Q} R) \ar[d] \\ H_{∗}(⋃ A) \ar[r]^{i_{∗}} & H_{∗}(X)}

commutes. It follows that i_{∗}(ϕ(x)) = 0. From this it follows that ϕ(x) ∈ im (H_{∗}(X, ⋃ A) ⟶^{𝔡} H_{∗}(⋃ A)) and therefore ϕ(x) • ϕ(y) = 0, since ϕ(x) • ϕ(y) ∈

im (H_{∗}(⋃ A, ⋃ A) ⟶^{𝔡} H_{∗}(⋃ A)).

From the commutativity of (1.5), it follows that

ϕ_{k}([a ⊗ ⟨p⟩ ⊗ m]) • ϕ_{l}([b ⊗ ⟨q⟩ ⊗ m′]) = mi_{∗}^{p}(a) • m′i_{∗}^{q}(b) = = mm′i_{∗}^{p∧q}(a • b) = ϕ_{k+l−n}([(a • b) ⊗ ⟨p ∧ q⟩ ⊗ mm′]),

where i^{p}: D(p) → X is the inclusion map. □

The intersection product in H_{∗}(X, ⋃ A) is much more interesting and will be our object of study in concrete classes of arrangements.

Chapter 2

Linear and related arrangements

Let V be an (n + 1)–dimensional vector space over 𝕂 with 𝕂 = ℂ or 𝕂 = ℝ. We consider a linear arrangement A in V , that is a finite set of proper linear subspaces of V . We define Q to be the intersection poset of A, that is the set {⋂ C: C ⊂ A} ordered by inclusion.

2.0.23 Definition. We denote the category of finite dimensional vector spaces over 𝕂 and linear monomorphisms by 𝔙. We define D^{A} ∈ 𝔙^{Q^{o}} by D^{Q}(q) := q and letting D^{A}(p ← q): D^{A}(p) → D^{A}(q) be the inclusion map.

2.0.24 Notation. For u ∈ Q, we set d(u) := dim_{𝕂} u − 1. For S ⊂ ℕ, we set Q_{S} := {q ∈ Q: d(q) ∈ S}.

2.1 Homology and Homotopy

ZŽ–maps

We will subsequently construct ZŽ–maps involving arrangements associated with the linear arrangement A. As a basis for these we will now construct a set of maps that could be called a linear ZŽ–map. All of these constructions will depend on choices of points in V with certain properties. The most fundamental case is the following.

2.1.1 Proposition. There is a function x assigning to each u ∈ Q a system (x_{j}^{u})_{0≤j≤d(u)} of d(u)+1 vectors in u such that for all k ∈ ℕ, u_{0}, …, u_{k} ∈ Q, with u_{0} < u_{1} < ⋯ < u_{r} and λ = (λ_{0}, …, λ_{r}) ∈ Δ^{r}, the system of vectors

(∑_{i=0}^{r} λ_{i}x_{j}^{u_{i}})_{0≤j≤d(u_{0})}

is linearly independent.

Proof. We give a simple recursive construction, because similar ones will be important later on. Let Λ be a linear functional on V that vanishes on no element of Q_{[0,n]} and set H := ker Λ. By induction there are (x_{j}^{u})_{0≤j<d(u)} with x_{j}^{u} ∈ u ∩ H such that for all k, r ∈ ℕ, u_{0} < u_{1} < ⋯ < u_{r}, λ ∈ Δ^{r}, the system (∑_{i} λ_{i}x_{j}^{u_{i}})_{0≤j<d(u_{0})} is linearly independent. (The case u_{0} = 0 is trivial.) We now

choose x_{d(u)}^{u} ∈ u \ H for all u. Now, if μ ∈ 𝕂^{d(u_{0})+1} with ∑_{j} μ_{j} ∑_{i} λ_{i}x_{j}^{u_{i}} = 0, then μ_{d(u_{0})}λ_{0}x_{d(u_{0})}^{u_{0}} ∈ H and therefore μ_{d(u_{0})}λ_{0} = 0. If λ_{0} = 0 then μ = 0 by

linear independence of (∑_{i=1}^{r} λ_{i}x_{j}^{u_{i}})_{0≤j≤d(u_{0})}. If μ_{d(u)} = 0, then μ = 0 by linear

independence of (∑_{i=0}^{r} λ_{i}x_{j}^{u_{i}})_{0≤j<d(u_{0})}. □

It will be important whether the space of possible choices of points is connected. One such case is the following.

2.1.2 Proposition. For 𝕂 = ℂ, the set of all functions x as in Proposition 2.1.1, considered as a subspace of the affine space ∏_{u_{∈}Q} u^{d(u)+1} contains a non-empty Zariski-open set and is hence path-connected.

Proof. The complement of the set is contained in the union of the sets

{x: Ex. 0 ≠ λ ∈ ℂ^{r+1} and 0 ≠ μ ∈ ℂ^{d(u_{0})+1} s.t. ∑_{i=0}^{k} ∑_{j=0}^{d(u_{0})} λ_{i}μ_{j}x_{j}^{u_{i}} = 0}

for all chains u_{0} < u_{1} < ⋯ < u_{r}. Since the defining equations are homogenous in λ and μ, these sets are algebraic by the main theorem of elimination theory[Sha94, I.5, Thm 3]. Since the affine space is irreducible, it will be sufficient to show that each of these sets has non-empty complement. So we fix a chain u_{0} < u_{1} < ⋯ < u_{k}. We choose a basis (e_{l})_{l=0,...,n} of V such that e_{l} ∈ u_{i} for l ≤ d(u_{0}) + l and set x_{j}^{i} := e_{i+j}. Now if ∑_{j} ∑_{i} μ_{j}λ_{i}x_{j}^{i} = 0 then ∑_{i} λ_{i}μ_{s−i} = 0 for all s, and it follows that λ = 0, μ = 0. □

2.1.3 Definition. Given a function x as in Proposition 2.1.1, we define functions f^{p} for p ∈ Q by

\begin{matrix}f^{p}: 𝕂^{d(p)+1} × Δ[p, V] & → V, \\ ((μ_{0}, …, μ_{d(p)}), ∑_{i=0}^{r} λ_{i}u_{i}) & ↦ ∑_{j=0}^{d(p)} ∑_{i=0}^{r} μ_{j}λ_{i}x_{j}^{u_{i}}. \end{matrix}  (2.1)

2.1.4 Proposition. For any y ∈ Δ[p, q], the map f^{p}(•, y) is a linear injection into q ⊂ V (and so necessarily an isomorphism, if p = q), and for p ≤ q, the maps f^{p} and f^{q} agree on 𝕂^{d(p)+1} × Δ[q, V] where we identify 𝕂^{d(p)+1} with {μ ∈ 𝕂^{d(q)+1} : μ_{i} = 0 for i > d(p)}. □

This proposition can be formulated as the maps f^{p} forming a commutative diagram as (1.6) with V for X and the functor F^{Q} now to be defined for E. With this definition we follow [WZŽ99, p. 141]

2.1.5 Definition. We define F^{Q} ∈ 𝔙^{Q^{o}} by F^{Q}(q) := 𝕂^{d(q)+1} and letting F^{Q}(p ← q): 𝕂^{d(p)+1} → 𝕂^{d(q)+1} be the standard inclusion.

Spherical arrangements

We now define and study the spherical arrangement 𝕊A associated to the linear arrangement A. This is done in some detail mainly to demonstrate the methods at our hands. The homological results will be obtained again later on as special cases of affine arrangements.

2.1.6 Definition. For a finite dimensional 𝕂–vector space u, we set 𝕊(u) := (u \ {0})/ ∼, where ∼ is the equivalence relation identifying x and y if there exists λ > 0 with x = λy. If u is equipped with a norm, we can alternatively set 𝕊(u) := {x ∈ u: |x| = 1}. For a linear injection f: u → v we define 𝕊(f): 𝕊(u) → 𝕊(v) by [x] ↦ [f(x)], making 𝕊 into a functor from 𝔙 to 𝔗op.

2.1.7 Definition. We define the spherical arrangement 𝕊A derived from A by 𝕊A := {𝕊A: A ∈ A}. The union ⋃ 𝕊A of 𝕊A is usually called the link of A and the complement 𝕊V \ ⋃ 𝕊A is homotopy equivalent the the complement V \ ⋃ A of A.

We will describe the homotopy type and the homology groups of the link of A, as well as the cohomology groups of the complement, which are isomorphic to H_{∗} (𝕊V, ⋃ 𝕊A) via Poincaré duality.

2.1.8 Definition. From the functions f^{p}: 𝕂^{d(p)+1} × Δ[p, V] → V we derive functions 𝕊(f^{p}): 𝕊(𝕂^{d(p)+1}) × Δ[p, V] → 𝕊(V) by 𝕊(f^{p})(x, y) := 𝕊(f^{p}(•, y))(x). More explicitly

\begin{matrix}𝕊(f^{p}): 𝕊(𝕂^{d(p)+1}) × Δ[p, V] & → 𝕊(V) \\ ([(μ_{0}, …, μ_{d(p)})], ∑_{i=0}^{r} λ_{i}u_{i}) & ↦ [∑_{j=0}^{d(p)} ∑_{i=0}^{r} μ_{j}λ_{i}x_{j}^{u_{i}} ] . \end{matrix}  (2.2)

We note that 𝕊(ℝ^{d(p)+1}) ≈ 𝕊^{d(p)} and 𝕊(ℂ^{d(p)+1}) ≈ 𝕊^{2d(p)+1}.

2.1.9 Proposition. The maps 𝕊(f^{p}) induce a homotopy equivalence

hcolim 𝕊(F^{Q_{[0,n)}}) ⟶^{≃} ⋃ 𝕊A.

Proof. By Proposition 2.1.4 the conditions of Proposition 1.3.5 are satisfied. □

This shows that the homotopy type of ⋃ 𝕊A depends on the intersection poset Q equipped with the dimension function d only. However, a different description of this homotopy type is more usual.

2.1.10 Proposition. Let M ∈ 𝔗op^{Q^{o}^{[0,n)}} be defined by M(q) := 𝕊(𝕂^{d(q)+1}) and all non-identity morphisms being constant to the base-point. Then hcolim M ≃ ⋃ 𝕊A.

Proof. We want to compare the diagrams M and 𝕊(F^{Q_{[0,n)}}). We have M(q) = 𝕊(F^{Q_{[0,n)}})(q) for all q, and the inclusion maps in 𝕊(F^{Q_{[0,n)}}) are homotopic to the constant maps in M. These homotopies can be arranged to form a ZŽ–map.

Let H: 𝕊^{∞} × I → 𝕊^{∞} be a base-point preserving homotopy from the identity to the constant map satisfying H[𝕊^{k} × I] ⊂ 𝕊^{k+1} for all k. The latter condition can be achieved for example by cellular approximation. We define maps f_{p}^{q} for p ≤ q, p, q ∈ Q_{[0,n)} by

\begin{matrix}f_{p}^{q}: M(p) × Δ[p, q] & → 𝕊(F^{Q_{[0,n)}})(q) \\ (x, ∑_{i=0}^{r} λ_{i}u_{i}) & ↦ \begin{cases}H(x, 1 − λ_{0}), & u_{0} = p, \\ ∗, & u_{0} > p. \end{cases} \end{matrix}

For p < p′, we have f_{p′}^{q}(M(p ← p′)(x), t) = H(∗, t) = ∗ = H(x, 1) = f_{p}^{q}(x, t).

For q < q′, f_{p}^{q′}(x, t) = F^{Q}(f_{p}^{q}(x, t)). Therefore, the maps f_{p}^{q} induce a map of Q^{o}_{[0,n)}-diagrams M ×_{Q} EQ → F^{Q_{[0,n)}} . Since f_{p}^{p} is a homeomorphism, this map of diagrams is a ZŽ–map. By Proposition 1.1.20 it induces a homotopy equivalence hcolim M ⟶^{≃} hcolim 𝕊(F^{Q_{[0,n)}}). □

2.1.11 Remark. Of course, it would have been just as easy to give a ZŽ–map M ×_{Q} EQ → 𝕊(D^{A}) directly.

We turn to homology calculations.

2.1.12 Proposition. The map

\begin{matrix}S(𝕊(F^{Q})) ⊗_{Q} B(Q) & → S(𝕊(D^{A})) \\ c × (p ← q_{0} ← … ← q_{n} ← p′) & ↦ 𝕊(f^{p})_{∗}(c × ⟨q_{0}, …, q_{n}⟩) \end{matrix}  (2.3)

is a ZŽ–map and therefore induces isomorphisms

H(S(𝕊(F^{Q})) ⊗_{Q} B(Q) ⊗_{Q} K^{u}) ⟶^{≅} H_{∗} (⋃ 𝕊A)  (2.4)

and

H(S(𝕊(F^{Q})) ⊗_{Q} B(Q) ⊗_{Q} K^{p}) ⟶^{≅} H_{∗} (𝕊V, ⋃ 𝕊A) .  (2.5)

Proof. This is an application of Proposition 1.3.12. The conditions of that Proposition are met because of Proposition 2.1.4. □

This argument is not specific to the functor 𝕊: 𝔙 → 𝔗op and we will investigate similar functors later on.

Proposition 2.1.12 already describes the homology of the link and, via Poincaré duality, the cohomology of the complement in terms of the intersection poset Q and the dimension function d. We will now simplify this description.

2.1.13 Definition and Proposition. Let e_{n} = [c_{n}] be a generator of H̃_{n}(𝕊^{n}) and b_{n} ∈ S_{n+1}(𝕊^{n+1}) with 𝔡b_{n} = c_{n}. Also let a := ⟨1⟩ ∈ C_{0}(S^{0}). The map

H(S(𝕊(F^{Q}))) ⊗_{Q} B(Q) → S(𝕊(F^{Q}))
[a] ⊗ (p ← q_{0} ← … ← q_{n} ← p′) ↦ \begin{cases}a, & n = 0 \\ 0, & n > 0 \end{cases}

e_{k} ⊗ (p ← q_{0} ← … ← q_{n} ← p′) ↦ \begin{cases}c_{k}, & n = 0, p = q_{0} \\ (−1)^{k+1}b_{k}, & n = 1, p = q_{0} < q_{1} \\ 0, & otherwise \end{cases}

(2.6)

is a ZŽ–map.

Proof. The map h_{r}: H(𝕊(𝕂^{r+1})) → S(𝕊(𝕂^{r+1})) mapping e_{k} to c_{k} and [a] to a is a chain homotopy equivalence. If r < s and i: 𝕊(𝕂^{r+1}) → 𝕊(𝕂^{s+1}) is the inclusion map, then e_{k} ↦ b_{k} is a chain homotopy from h_{s} ∘ H(i) to S(i) ∘ h_{r}. The map (2.6) can now be seen as a special case of that from Proposition 1.2.13 □

2.1.14 Proposition. There are isomorphisms

H(H(S(𝕊(F^{Q}))) ⊗_{Q} B(Q) ⊗_{Q} K^{u}) ⟶^{≅} H_{∗} (⋃ 𝕊A)  (2.7)

and

H(H(S(𝕊(F^{Q}))) ⊗_{Q} B(Q) ⊗_{Q} K^{p}) ⟶^{≅} H_{∗} (𝕊V, ⋃ 𝕊A) .  (2.8)

Proof. By Proposition 1.2.15, the ZŽ–maps from Proposition 2.1.12 and (2.6) combine to yield ZŽ–maps inducing the desired isomorphisms. □

2.1.15 Proposition. For 𝕂 = ℝ the above maps induce isomorphisms

H_{∗} (⋃ 𝕊A) ≅ H_{∗}(Δ(Q_{[0,n)})) ⊕ ⨁_{q∈Q_{[0,n)}} H_{∗}(Δ[q, V), Δ(q, V))[−d(q)],

H_{∗} (𝕊V, ⋃ 𝕊A) ≅ H_{∗}(ΔQ_{[0,n]}, ΔQ_{[0,n)})

⊕ ⨁_{q∈Q_{[0,n]}} H_{∗}(Δ[q, V], Δ[q, V) ∪ Δ(q, V])[−d(q)].

Proof. The Q^{o}–diagram H(𝕊(F^{Q})) decomposes as

H(𝕊(F^{Q})) = P ⊕ ⨁_{q_{∈}Q_{[0,n]}} O_{q}  (2.9)

with P corresponding to the class of a 0-simplex and O_{q} to a generator of H̃_{d(q)}(𝕊^{d(q)}), i.e.

P (q) ≅ \begin{cases}R, & 0 ≤ d(q) ≤ n, \\ 0, & d(q) = −1, \end{cases}  (2.10)

and all morphism the identity where possible, and

O_{q}(p) ≅ \begin{cases}R[−d(q)], & p = q, \\ 0, & p ≠ q. \end{cases}  (2.11)

Now

H(P ⊗_{Q} B(Q) ⊗_{Q} K^{u}) ≅ H_{∗}(ΔQ_{[0,n)}),
H(P ⊗_{Q} B(Q) ⊗_{Q} K^{p}) ≅ H_{∗}(ΔQ_{[0,n]}, ΔQ_{[0,n)}),
H(Q_{q} ⊗_{Q} B(Q) ⊗_{Q} K^{u}) ≅ H_{∗}(Δ[q, V), Δ(q, V))[−d(q)],
H(Q_{q} ⊗_{Q} B(Q) ⊗_{Q} K^{p}) ≅ H_{∗}(Δ[q, V], Δ[q, V) ∪ Δ(q, V])[−d(q)]

proving the proposition. □

2.1.16 Proposition. For 𝕂 = ℝ

H̃_{∗} (⋃ 𝕊A) ≅ ⨁_{q∈[⊥,V)} H̃_{∗}(Δ(q, V))[−(d(q) + 1)],  (2.12)
H_{∗} (𝕊V, ⋃ 𝕊A) ≅ ℤ[−n] ⊕ ⨁_{q∈[⊥,V)} H̃_{∗}(Δ(q, V))[−d(q) − 2].  (2.13)

Proof. Since [q, V) has, for q < V , the minimum q, Δ[q, V) is acyclic and H_{k}(Δ[q, V), Δ(q, V)) ≅ H̃_{k−1}(Δ(q, V)). If ⊥ ∈ Q_{[0,n)}, then Q_{[0,n)} has ⊥ as its minimum and H̃_{∗}(ΔQ_{[0,n)}) = 0. Otherwise, H̃_{∗}(ΔQ_{[0,n)}) = H̃_{∗}(Δ(⊥, V)). This proves the first isomorphism.

H_{k}(ΔQ_{[0,n]}, ΔQ_{[0,n)}) ≅ H̃_{k−1}(Δ_{[0,n)}) and for q < V

H_{k}(Δ[q, V], Δ[q, V) ∪ Δ(q, V]) ≅ H_{k−1}(Δ[q, V), Δ(q, V)),

and these have been dealt with in the preceding paragraph. Also

H_{∗}(Δ[V, V], Δ[V, V) ∪ Δ(V, V])[−d(V)] = H_{∗}(Δ {V })[−n] ≅ ℤ[−n],

completing the proof of the second isomorphism. □

Projective arrangements

We now come to our main object of study, projective arrangements.

For the functor P associating to a 𝕂–vector–space u the projective space P u we proceed just as for the functor 𝕊 before.

2.1.17 Proposition. The maps P (f^{p}) induce a homotopy equivalence

hcolim P (F^{Q_{[0,n)}}) ⟶^{≃} ⋃ P A.

Proof. Again, by Proposition 2.1.4 the conditions of Proposition 1.3.5 are satisfied. □

2.1.18 Remark. This is the same description of the homotopy type of a projective arrangement as in [WZŽ99, Prop. 5.9]. Our methods are very similar and also applicable to the other cases considered there. Our approach which makes ZŽ–maps more central has the advantage of explicitly constructing a homotopy equivalence.

2.1.19 Proposition. The map

\begin{matrix}S(P (F^{Q})) ⊗_{Q} B(Q) & → S(P (D^{A})) \\ c × (p ← q_{0} ← … ← q_{n} ← p′) & ↦ P (f^{p})_{∗}(c × ⟨q_{0}, …, q_{n}⟩) \end{matrix}  (2.14)

is a ZŽ–map and therefore induces isomorphisms

H(S(P (F^{Q})) ⊗_{Q} B(Q) ⊗_{Q} K^{u}) ⟶^{≅} H_{∗} (⋃ P A)  (2.15)

and

H(S(P (F^{Q})) ⊗_{Q} B(Q) ⊗_{Q} K^{p}) ⟶^{≅} H_{∗} (P V, ⋃ P A) .  (2.16)

Proof. Just as Proposition 2.1.12 □

We simplify this description, starting with the most difficult case. The following result will actually also hold in the other cases treated afterwards.

2.1.20 Proposition. Let 𝕂 = ℝ and R = ℤ. Then

H_{∗} (⋃ P A) ≅ H(H(P (F^{Q})) ⊗_{Q} B(Q) ⊗_{Q} K^{u})  (2.17)

and

H_{∗} (P V, ⋃ P A) ≅ H(H(P (F^{Q})) ⊗_{Q} B(Q) ⊗_{Q} K^{p}).  (2.18)

Proof. Let M ∈ 𝔡𝔄b^{Q^{o}} be defined by

M(q)_{k} = \begin{cases}ℤ, & 0 ≤ k ≤ d(q) \\ 0, & otherwise, \end{cases} M(q)_{k+1} ⟶^{𝔡} M(q)_{k} = \begin{cases}2, & 0 ≤ k < d(q), k odd, \\ 0, & otherwise, \end{cases}

and the morphisms in M the identity where possible. It follows from the usual cell decomposition of ℝP^{k} with one cell per dimension that there exists a chain map of Q^{o}-diagrams M → F^{Q} inducing an isomorphism in homology. M allows a direct sum decomposition M = ⨁_{i} M^{i} with M_{k}^{i} := M_{k} for k ∈ {2i − 1, 2i} and M_{k}^{i} := 0 otherwise. The homology of M^{i} is nonzero in a single dimension only, hence Proposition 1.2.18 is applicable and yields the result together with Proposition 2.1.19. □

Possibly looking less attractive, these are as explicit combinatorial formulas as those derived in Proposition 2.1.24 for other cases. Nevertheless, for calculating the groups in question, the following may be helpful.

2.1.21 Proposition. There are isomorphisms

H(H_{0}(P (F^{Q})) ⊗_{Q} B(Q) ⊗_{Q} K^{u}) ≅ H_{∗}(ΔQ_{[0,n)}),
H(H_{2i}(P (F^{Q})) ⊗_{Q} B(Q) ⊗_{Q} K^{u}) ≅ 0 for i > 0,

and there is a long exact sequence

⋯ → H_{k}(ΔQ_{[2i+2,n)}) ⟶^{2j_{∗}} H_{k}(ΔQ_{[2i+1,n)})

→ H_{k}(H_{2i+1}(P (F^{Q})) ⊗_{Q} B(Q) ⊗_{Q} K^{u}) → H_{k−1}(ΔQ_{[2i+2,n)}) ⟶^{2j_{∗}} ⋯

for i ≥ 0, where j is the inclusion map. Similarly, there are isomorphisms

H(H_{0}(P (F^{Q})) ⊗_{Q} B(Q) ⊗_{Q} K^{p}) ≅ H_{∗}(ΔQ_{[0,n]}, ΔQ_{[0,n)}),
H(H_{2i}(P (F^{Q})) ⊗_{Q} B(Q) ⊗_{Q} K^{p}) ≅ 0 for i > 0,

and there is a long exact sequence

⋯ → H_{k}(ΔQ_{[2i+2,n]}, ΔQ_{[2i+2,n)}) ⟶^{2j_{∗}} H_{k}(ΔQ_{[2i+1,n]}, ΔQ_{[2i+1,n)}) →

H_{k}(H_{2i+1}(P (F^{Q})) ⊗_{Q} B(Q) ⊗_{Q} K^{p}) → H_{k−1}(ΔQ_{[2i+2,n]}, ΔQ_{[2i+2,n)}) ⟶^{2j_{∗}} ⋯

for i ≥ 0.

Proof. The parts dealing with H(H_{2i+1}(P (F^{Q})) ⊗_{Q} B(Q) ⊗_{Q} K) for K = K^{u} or K = K^{p} are the interesting ones. With M_{k} ∈ 𝔄b^{Q^{o}} as in the preceding proof, there is a short exact sequence 0 → M_{2i+2} ⟶^{2} M_{2i+1} → H_{2i+1}(P (F^{Q})) → 0. Since B(Q) ⊗_{Q} K is free, this induces a short exact sequence

0 → M_{2i+2} ⊗_{Q} B(Q) ⊗_{Q} K ⟶^{2} M_{2i+1} ⊗_{Q} B(Q) ⊗_{Q} K

→ H_{2i+1}(P (F^{Q})) ⊗_{Q} B(Q) ⊗_{Q} K → 0

and the long exact sequences stated in the proposition. □

The following cases are easier.

2.1.22 Definition and Proposition. Let 𝕂 = ℂ, or let 𝕂 = ℝ and R = ℤ_{2}. Let e_{n} = [c_{n}] be a generator of H_{2n}(ℂP^{n}) or H_{n}(ℝP^{n}; ℤ_{2}) respectively. For i ≥ n we define e_{n}^{i} ∈ H_{∗}(𝕂P^{i}; R) by e_{n}^{i} = [c_{n}]. The maps

\begin{matrix}H(P (F^{Q}(u))) & → S(P (F^{Q}(u))), \\ e_{n}^{i} & ↦ c_{n} \end{matrix}  (2.19)

for u ∈ Q induce isomorphisms in homology and form a chain map of Q^{o}– diagrams H(S(P (F^{Q}))) → S(P (F^{Q})). □

2.1.23 Proposition. For 𝕂 = ℂ, as well as for 𝕂 = ℝ and R = ℤ_{2}, there are isomorphisms

H(H(P (F^{Q})) ⊗_{Q} B(Q) ⊗_{Q} K^{u}) ⟶^{≅} H_{∗} (⋃ P A)  (2.20)

and

H(H(P (F^{Q})) ⊗_{Q} B(Q) ⊗_{Q} K^{p}) ⟶^{≅} H_{∗} (P V, ⋃ P A) .  (2.21)

Proof. The composition of (2.19) with the ZŽ–map (2.14) again yields a ZŽ–map and can therefore be substituted for (2.14) in Proposition 2.1.19. □

2.1.24 Proposition. The above maps induce isomorphisms

H_{∗} (⋃ P A; ℤ_{2}) ≅ ⨁_{k=0}^{n} H_{∗}(ΔQ_{[k,n)}; ℤ_{2})[−k],  (2.22)

H_{∗} (P V, ⋃ P A; ℤ_{2}) ≅ ⨁_{k=0}^{n} H_{∗}(ΔQ_{[k,n]}, ΔQ_{[k,n)}; ℤ_{2})[−k]  (2.23)

for 𝕂 = ℝ, and

H_{∗} (⋃ P A) ≅ ⨁_{k=0}^{n} H_{∗}(ΔQ_{[k,n)})[−2k],  (2.24)

H_{∗} (P V, ⋃ P A) ≅ ⨁_{k=0}^{n} H_{∗}(ΔQ_{[k,n]}, ΔQ_{[k,n)})[−2k]  (2.25)

for 𝕂 = ℂ.

Proof. For 𝕂 = ℂ or 𝕂 = ℝ and R = ℤ_{2}, there is a direct sum decomposition

H(P (F^{Q})) = ⨁_{k=0}^{n} O_{k},  O_{k}(q) ≅ \begin{cases}R[−mk], & d(q) ≥ k, \\ 0, & d(q) < k \end{cases}  (2.26)

with m = dim_{ℝ} 𝕂, and O_{k}(q ← q′) = id for d(q) ≥ k. We choose the isomorphism in such a way that 1 ∈ O_{k}(q) corresponds to the canonical generator of H_{mk}(𝕂P^{d(q)}). Now

H(O_{k} ⊗_{Q} B(Q) ⊗_{Q} K^{u}) ≅ H_{∗}(ΔQ_{[k,n)})[−mk],  (2.27)
H(O_{k} ⊗_{Q} B(Q) ⊗_{Q} K^{p}) ≅ H_{∗}(ΔQ_{[k,n]}, ΔQ_{[k,n)})[−mk],  (2.28)

and the result follows. □

We assemble the different maps to get the more direct handle on these isomorphisms that we will need when discussing intersection products. For Section 2.3 this, together with the preceding proposition, is the main result of this section.

2.1.25 Definition and Proposition. We define

\begin{matrix}f^{k}: 𝕂P^{k} × (ΔQ_{[k,n]}, ΔQ_{[k,n)}) & → (P V, ⋃ P A) \\ ([(μ_{0}, …, μ_{k})], ∑_{i=0}^{r} λ_{i}u_{i}) & ↦ [∑_{j=0}^{k} ∑_{i=0}^{r} μ_{j}λ_{i}x_{j}^{u_{i}} ] \end{matrix}  (2.29)

and

\begin{aligned}h_{k}: H_{∗}(ΔQ_{[k,n]}, ΔQ_{[k,n)}) & → H_{∗} (P V, ⋃ P A) , \\ H_{∗}(ΔQ_{[k,n)}) & → H_{∗} (⋃ P A) \\ c & ↦ f_{∗}^{k}([𝕂P^{k}] × c), \end{aligned}  (2.30)

where [𝕂P^{k}] is the orientation class of 𝕂P^{k}, over ℤ_{2} in case 𝕂 = ℝ. Then all of the isomorphisms from Proposition 2.1.24 are given by ∑_{k=0}^{n} h_{k}.

For 𝕂 = ℂ, the maps h_{k} do not depend on the choice of the function x.

Proof. It is trivial to check that ∑_{k} h_{k} is indeed the composition of the maps (2.14), (2.19), and (2.26). The non-dependence on the choice of x follows from Proposition 2.1.2. □

2.1.26 Remark. The calculations regarding the homology of the projective arrangements could equally well been carried out using the power set of A instead of the intersection poset, i.e., in the words of Remark 1.3.8, using naive resolutions instead of economical ones. This is because the key fact that all maps in the diagram H(P (F^{Q})) are injective (assuming 𝕂 = ℂ or R = ℤ_{2}) still holds when Q denotes the power set of A. For spherical arrangements however, naive resolutions would have been less appropriate, because the fact that all non-identity homomorphisms in the diagram H_{k}(𝕊(F^{Q})) are zero for k > 0 would be lost, because there are non-identity isomorphisms in F^{Q} when Q denotes the power set of A, so extra care has to be taken.

Affine arrangements

Let H be a hyperplane in V . We will investigate the relationship between the projective arrangement P A, the projective arrangement induced on P H, and the arrangement induced on the affine space P V \ P H.

We set A^{H} := {A ∩ H: A ∈ A} and denote the intersection poset of A^{H} by Q^{H}.

We also set Q̄ := Q \ Q^{H} = {u ∈ Q: u ⊄ H}. This is the poset of non-empty intersections of the affine arrangement in P V \ P H.

2.1.27 Definition. We call the arrangement A a ≥ 2–arrangement, if u < v implies d(v) − d(u) ≥ 2 for all u, v ∈ Q.

2.1.28 Notation. Let X be a compact m-manifold and A ⊂ X a closed subset. If X \ A is orientable and such an orientation is chosen, we denote by [X, A] the corresponding orientation class in H_{m}(X, A).

2.1.29 Definition and Proposition. We consider for u ∈ Q̄ systems of vectors b_{0}^{u}, …, b_{d(u)−1}^{u} ∈ u ∩ H, x_{u}^{u} ∈ u, x_{v}^{u} ∈ v ∩ H for v > u, such that

\begin{matrix}f^{u}: 𝕂P^{d(u)} × Δ[u, V] & → P V \\ ([μ_{0} : ⋯: μ_{d(u)}], ∑_{j=0}^{r} λ_{j}v_{j}) & ↦ [ ∑_{i=0}^{d(u)−1} μ_{i}b_{i}^{u} + μ_{d(u)} ∑_{j=0}^{r} λ_{j}x_{v_{j}}^{u} ] \end{matrix}  (2.31)

is well defined, i.e. the term on the right hand side never equals zero. In particular the system of vectors b_{0}^{u}, …, b_{d(u)−1}^{u}, x_{u}^{u} determines, in case 𝕂 = ℝ, an orientation of (P u, P (u ∩ H)) via the homeomorphism

\begin{matrix}f̄^{u}: (𝕂P^{d(u)}, 𝕂P^{d(u)−1}) & → (P u, P (u ∩ H)) \\ [μ_{0} : ⋯: μ_{d(u)}] & ↦ [ ∑_{i=0}^{d(u)−1} μ_{i}b_{i}^{u} + μ_{d(u)}x_{u}^{u}] \end{matrix}  (2.32)

which is obtained by restricting f^{u}.

Such vectors exist, and the induced orientation can be prescribed in case 𝕂 = ℝ. There are induced maps

\begin{aligned}h_{u}: H_{∗}(Δ[u, V], Δ[u, V) ∪ Δ(u, V]) & → H_{∗} (P V, P H ∪ ⋃ P A) , \\ H_{∗}(Δ[u, V), Δ(u, V)) & → H_{∗} (P V, P H ∪ ⋃ P A) , \\ c & ↦ f_{∗}^{u}([𝕂P^{d(u)}, 𝕂P^{d(u)−1}] × c). \end{aligned}  (2.33)

For 𝕂 = ℂ these are independent of the choice of b^{u} and x^{u}. For 𝕂 = ℝ and if A is a ≥ 2–arrangement, they depend only on the induced orientation of (P u, P (u ∩ H)). The maps

⨁_{u∈Q̄} H_{∗}(Δ[u, V], Δ[u, V) ∪ Δ(u, V])[−md(u)] \underrightarrow{∑_{u} h_{u}} H_{∗} (P V, P H ∪ ⋃ P A) , (2.34)

⨁_{u∈Q̄} H_{∗}(Δ[u, V), Δ(u, V))[−md(u)] \underrightarrow{∑_{u} h_{u}} H_{∗} (⋃ P A, P H ∩ ⋃ P A) (2.35)

with m = dim_{ℝ} 𝕂 are isomorphisms.

2.1.30 Remark. If d(⊥) ≥ 0, that is if ⊥ ∈ Q̄, the affine arrangement is a (central) linear arrangement and ⋃ P A/ (P H ∩ ⋃ P A) is homeomorphic to the suspension of the link of that arrangement. Therefore, these isomorphisms generalize those of Proposition 2.1.15. The proof of Proposition 2.1.16 explains how to deal with the first summands in Proposition 2.1.15 when comparing it with the current proposition.

Proof. For the vectors b_{i}^{u} and x_{v}^{u} to define a map f^{u}, it suffices that for every chain u ≤ v_{0} < ⋯ < v_{r} the vectors b_{0}^{u}, …, b_{d(u)−1}^{u}, x_{v_{0}}^{u}, …, x_{v_{r}}^{u} are linearly independent. This will be the case, if (b_{i}^{u})_{i} is a basis of u ∩ H, x_{u}^{u} ∈ u \ H and x_{v}^{u} ∈ v \ w for u ≤ w < v. The systems of vectors with this property form a nonempty Zariski-open set, which is therefore dense in the set of all allowed systems, and it is path-connected for 𝕂 = ℂ. For 𝕂 = ℝ and if A is a ≥ 2-arrangement, it has at most four components, distinguished by the orientation of u ∩ H that (b_{i}^{u})_{i} defines and the component of u \ H that contains x_{u}^{u}. Since replacing all of the b_{i}^{u} and x_{v}^{u} by their negatives does not change f^{u}, we may restrict x_{u}^{u} to one of the components of u \ H, and we see that the homotopy type of f^{u} depends only on the orientation induced on (P u, P (u ∩ H)).

We set A(u) := (P u, P (u ∩ H)) for u ∈ Q̄. By Proposition 1.3.13 there are isomorphisms

H(S(A(D̄^{A})) ⊗Q̄ K^{p}) ⟶^{≅} H_{∗} (P V, P H ∪ ⋃ P A)  (2.36)

and

H(S(A(D̄^{A})) ⊗Q̄ K^{u}) ⟶^{≅} H_{∗} (⋃ P A, P H ∩ ⋃ P A) .  (2.37)

Let O_{u} ∈ 𝔡𝔄bQ̄^{o} be defined by

O_{u}(q) := \begin{cases}R[−md(u)], & u = q, \\ 0, & u ≠ q \end{cases}

and g_{u} ∈ Hom_{ℭ^{o}}(O_{u} ⊗Q̄ B(Q̄), S(A(D̄^{A}))) by

1_{R} ⊗ (u ← v_{0} ← ⋯ ← v_{r} ← u′) ↦ f_{∗}^{u}(o_{d(u)} × ⟨v_{0}, …, v_{r}⟩),

where o_{k} is a relative cycle representing the orientation class of (𝕂P^{k}, 𝕂P^{k−1}). This is well defined, because f_{∗}^{u}(o_{d(u)} × ⟨v_{0}, …, v_{r}⟩) = 0 for v_{0} > u. It is a chain map, because f_{∗}^{u}(𝔡o_{d(u)} × ⟨v_{0}, …, v_{r}⟩) = 0. We have

H(O_{u} ⊗Q̄ B(Q̄) ⊗Q̄ K^{p}) ≅ H_{∗}(Δ[u, V], Δ[u, V) ∪ Δ(u, V])[−md(u)],
H(O_{u} ⊗Q̄ B(Q̄) ⊗Q̄ K^{u}) ≅ H_{∗}(Δ[u, V), Δ(u, V))[−md(u)],

and the maps h_{u} equal H(g_{u} ⊗ id) composed with the above isomorphisms. The map ∑_{u}: ⨁_{u} O_{u} ⊗Q̄ B(Q̄) → S(A(D̄^{A})) is a ZŽ–map to a free diagram, therefore ∑_{u} h_{u} is an isomorphism. □

That the direct sum decomposition ⨁_{u} H_{∗}(Δ[u, V], Δ[u, V) ∪ Δ(u, V])[−md(u)] is finer than the decomposition ⨁_{k} H_{∗}(ΔQ_{[k,n]}, ΔQ_{[k,n)})[−mk] can be regarded as a reason why products in affine arrangements are technically simpler than products in projective arrangements. In particular, there seems to be no direct generalization of the proof of Proposition 2.2.12 to a proof of Proposition 2.3.13.

We illustrate the connection between projective and affine arrangements by giving another description of the homomorphisms h_{u}. For simplicity, we only cover the case 𝕂 = ℂ completely.

2.1.31 Definition and Proposition. There is a function x as in Proposition 2.1.1 with the additional property that x_{j}^{u} ∈ H for j < dim_{𝕂}(u ∩ H). If such a function is used in definition of the map f^{k} from Definition 2.1.25, then f^{k} maps the subspace 𝕂P^{k} × ΔQ_{(k,n]} ∪ 𝕂P^{k−1} × ΔQ_{[k,n]} to P H and therefore induces maps

\begin{aligned}h̄_{k}: H_{∗}(ΔQ̄_{[k,n]}, ΔQ̄_{[k,n)} ∪ ΔQ̄_{(k,n]}) & → H_{∗} (P V, P H ∪ ⋃ P A) , \\ H_{∗}(ΔQ̄_{[k,n)}, ΔQ̄_{(k,n)}) & → H_{∗} (⋃ P A, P H ∩ ⋃ P A) , \\ c & ↦ f_{∗}^{k}([𝕂P^{k}, 𝕂P^{k−1}] × c). \end{aligned}  (2.38)

Denoting the inclusion map Δ[u, V] → ΔQ_{[d(u),n]} by i^{u}, the maps

∑_{u∈Q̄_{{k}}} i_{∗}^{u}: ⨁_{u∈Q̄_{{k}}} H_{∗}(Δ[u, V], Δ[u, V) ∪ Δ(u, V])

⟶^{≅} H_{∗}(ΔQ̄_{[k,n]}, ΔQ̄_{[k,n)} ∪ ΔQ̄_{(k,n]}) (2.39)

and

∑_{u∈Q̄_{{k}}} i_{∗}^{u}: ⨁_{u∈Q̄_{{k}}} H_{∗}(Δ[u, V), Δ(u, V)) ⟶^{≅} H_{∗}(ΔQ̄_{[k,n)}, ΔQ̄_{(k,n)})  (2.40)

are isomorphisms.

For 𝕂 = ℂ we have h_{u} = h̄_{k} ∘ i_{∗}^{u} and therefore the maps

∑ h̄_{k}: ⨁_{k} H_{∗}(ΔQ̄_{[k,n]}, ΔQ̄_{[k,n)} ∪ ΔQ̄_{(k,n]})[−2k] → H_{∗} (P V, P H ∪ ⋃ P A) , (2.41)

⨁_{k} H_{∗}(ΔQ̄_{[k,n)}, ΔQ̄_{(k,n)})[−2k] → H_{∗} (⋃ P A, P H ∩ ⋃ P A) (2.42)

are isomorphisms.

Proof. Slightly modifying the construction in the proof of Proposition 2.1.1, we first choose (x_{j}^{u})_{0≤j<dim(u∩H)} (applying the proposition to A^{H}) and then choose x_{d(u)}^{u} ∈ u \ H for all u ∈ Q̄. As in that proof, x has the desired property.

The maps in (2.39) and (2.39) are easily seen to be isomorphisms either directly at the chain level or by a Mayer-Vietoris argument using ΔQ̄_{[k,n]} = ⋃_{u∈Q̄_{{k}}} Δ[u, V] and Δ[u, V] ∩ Δ[v, V] ⊂ ΔQ̄_{(k,n]} for u, v ∈ Q̄_{{k}}, u ≠ v.

Let u ∈ Q̄. Any system of points (y_{j}^{v}) for v ≥ u and 0 ≤ j ≤ d(u) defines maps

\begin{aligned}g: (𝕂P^{d(u)}, 𝕂P^{d(u)−1}) × (Δ[u, V], Δ[u, V) ∪ Δ(u, V]) & → (P V, P H ∪ ⋃ P A) \\ (𝕂P^{d(u)}, 𝕂P^{d(u)−1}) × (Δ[u, V), Δ(u, V)) & → (⋃ P A, P H ∩ ⋃ P A) \\ ([(μ_{0}, …, μ_{k})], ∑_{i=0}^{r} λ_{i}u_{i}) & ↦ [∑_{j=0}^{k} ∑_{i=0}^{r} μ_{j}λ_{i}x_{j}^{u_{i}} ] , \end{aligned}

if the right hand side is always well defined. For 𝕂 = ℂ the space of such systems of points is again path connected and hence the homotopy type of g independent of the choice of such a system.

Setting y_{j}^{v} := x_{j}^{v} gives such a system, and for this choice g = f^{k} ∘i^{u}. Setting y_{j}^{v} := b_{j}^{u} for j < d(u) and y_{d(u)}^{v} := x_{v}^{u} with b and x the systems from Definition 2.1.29 gives another such system, and for this one g = f^{u}. In case 𝕂 = ℂ it follows that f^{k} ∘ i^{u} ≃ f^{u} and h_{u} = h̄_{k} ∘ i_{∗}^{u}. □

2.1.32 Remark. For a complex affine arrangement the isomorphism

H_{∗} (P V, P H ∪ ⋃ P A) ≅ ⨁_{k} H_{∗}(ΔQ̄_{[k,n]}, ΔQ̄_{[k,n)} ∪ ΔQ̄_{(k,n]})[−2k],

and hence the isomorphism (2.34), can also be deduced combinatorially from the isomorphism (2.25) for projective arrangements by considering the arrangement A^{+} := A ∪ {H}. This is called an ‘interesting exercise’ in [GM88]. We sketch how to do this. To this end we denote the intersection poset of A^{+} by Q^{+}. For 0 ≤ k < n, the simplical complex Δ(Q_{[k,n)}^{+} \ Q̄_{{k}}) is acyclic, since it contains the cone Δ {q: q ≤ H, d(q) ≥ k} as a deformation retract. Therefore the first map in

H_{∗}(ΔQ_{[k,n]}^{+}, ΔQ_{[k,n)}^{+}) ⟶^{≅} H_{∗}(ΔQ_{[k,n]}^{+}, ΔQ_{[k,n)}^{+} ∪ Δ(Q_{[k,n]}^{+} \ Q̄_{{k}}))

⟵^{≅} H_{∗}(ΔQ̄_{[k,n]}, ΔQ̄_{[k,n)} ∪ ΔQ̄_{(k,n]}),

which is induced by inclusion, is an isomorphism. The second map is also induced by inclusion and is an isomorphism by excision. The isomorphisms also hold in the trivial case k = n. This yields H_{∗} (P V, P H ∪ ⋃ P A) = H_{∗} (P V, ⋃ P A^{+}) ≅ ⨁ H_{∗}(ΔQ_{[k,n]}^{+}, ΔQ_{[k,n)}^{+})[−2k] ≅ ⨁ H_{∗}(ΔQ̄_{[k,n]}, ΔQ̄_{[k,n)} ∪ ΔQ̄_{(k,n]})[−2k].

2.1.33 Remark. If the arrangement A is in general position with respect to H, then Q̄ = Q_{[0,n]} and q ↦ q ∩H induces isomorphisms η: Q_{k+1} → Q_{k}^{H}. For 𝕂 = ℂ it follows from the construction in Proposition 2.1.31 that the diagram

\xymatrix{H_{i}(ΔQ_{[k,n−1]}^{H}, ΔQ_{[k,n−1)}^{H}) \ar[r]^{h_{k}^{H}} \ar[d]^{(η^{−1})_{∗}} & H_{i+2k}(P H, ⋃ P A^{H}) \ar[d]^{incl_{∗}} \\ H_{i}(ΔQ_{[k,n]}, ΔQ_{[k,n)}) \ar[r]^{h_{k}} \ar[d]^{incl_{∗}} & H_{i+2k}(P V, ⋃ P A) \ar[d]^{incl_{∗}} \\ H_{i}(ΔQ̄_{[k,n]}, ΔQ̄_{(k,n]} ∪ ΔQ̄_{[k,n)}) \ar[r]^{h̄_{k}} & H_{i+2k}(P V, P H ∪ ⋃ P A)}

commutes, since f^{k} ∘ (id_{ℂP^{k}} × η^{−1}) is a suitable function f_{H}^{k} for the definition of h_{k}^{H}. Both columns are part of a long exact sequence of the form

H_{j}(B, A ∩ B) → H_{j}(X, B) → H_{j}(X, A ∪ B) ⟶^{𝔡} H_{j−1}(B, A ∩ B)

for an excisive triad (X; A, B). Because of the naturality of the connecting homomorphism, the diagram

\xymatrix{H_{i}(ΔQ̄_{[k,n]}, ΔQ̄_{(k,n]} ∪ ΔQ̄_{[k,n)}) \ar[r]^{h̄_{k}} \ar[d]^{η_{∗}∘𝔡} & H_{i+2k}(P V, P H ∪ ⋃ P A) \ar[d]^{𝔡} \\ H_{i−1}(ΔQ_{[k,n−1]}^{H}, ΔQ_{[k,n−1)}^{H}) \ar[r]^{h_{k}^{H}} & H_{i+2k−1}(P H, ⋃ P A^{H})}

also commutes. Analogous commutative diagrams exist for the long exact sequence of the pair (⋃ P A, ⋃ P A^{H}). If the arrangement A is not in general position with respect to H, then the same commutative diagrams exist, but the construction of the left column requires more care.

2.1.34 Remark (Gysin sequence). We continue the preceding remark. We set A := P V \ P H and denote by A^{A} := P B \ P H: B ∈ A the induced arrangement in this affine space. Using Poincaré duality, which we denote by D, we switch to cohomology and obtain the following commutative diagram with exact columns.

\xymatrix{⨁_{k≥0} H_{2(n−k−1)−i}(ΔQ_{[k,n−1]}^{H}, ΔQ_{[k,n−1)}^{H}) \ar[r]_{≅}^{D∘∑_{k} h_{k}^{H}} \ar[d]^{(η^{−1})_{∗}} & H^{i}(P H \ ⋃ P A^{H}) \ar[d]^{i^{!}} \\ ⨁_{k≥0} H_{2(n−k−1)−i}(ΔQ_{[k,n]}, ΔQ_{[k,n)}) \ar[r]_{≅}^{D∘∑_{k} h_{k}} \ar[d]^{incl_{∗}} & H^{i+2}(P V \ ⋃ P A) \ar[d]^{incl^{∗}} \\ ⨁_{k≥0} H_{2(n−k−1)−i}(ΔQ̄_{[k,n]}, ΔQ̄_{(k,n]} ∪ ΔQ̄_{[k,n)}) \ar[r]_{≅}^{D∘∑_{k} h̄_{k}} \ar[d]^{η_{∗}∘𝔡} & H^{i+2}(A \ ⋃ A^{A}) \ar[d] \\ ⨁_{k≥0} H_{2(n−k−1)−i−1}(ΔQ_{[k,n−1]}^{H}, ΔQ_{[k,n−1)}^{H}) \ar[r]_{≅}^{D∘∑_{k} h_{k}^{H}} & H^{i+1}(P H \ ⋃ P A^{H})}

If the arrangement A^{A} is central, i.e. ⋂ A^{A} ≠ Ø, then there is a deformation retraction π: P V \ ⋃ P A ⟶^{≃} P H \ ⋃ P A^{H} homotopy inverse to the inclusion i. Also Q̄ = Q. From the above we therefore obtain the following diagram.

\xymatrix{⨁_{k≥0} H_{2(n−k−1)−i}(ΔQ_{[k+1,n]}, ΔQ_{[k+1,n)}) \ar[r]_{≅}^{D∘∑ h_{k}^{H}∘η_{∗}} \ar[d]^{incl_{∗}} & H^{i}(P H \ ⋃ P A^{H}) \ar[d]^{⌣α} \\ ⨁_{k≥0} H_{2(n−k−1)−i}(ΔQ_{[k,n]}, ΔQ_{[k,n)}) \ar[r]_{≅}^{D∘i_{!}∘∑_{k} h_{k}} \ar[d]^{incl_{∗}} & H^{i+2}(P H \ ⋃ P A^{H}) \ar[d]^{π^{∗}} \\ ⨁_{k≥0} H_{2(n−k−1)−i}(ΔQ_{[k,n]}, ΔQ_{(k,n]} ∪ ΔQ_{[k,n)}) \ar[r]_{≅}^{D∘∑ h̄_{k}} \ar[d]^{𝔡} & H^{i+2}(A \ ⋃ A^{A}) \ar[d] \\ ⨁_{k≥0} H_{2(n−k−1)−i−1}(ΔQ_{[k+1,n]}, ΔQ_{[k+1,n)}) \ar[r]_{≅}^{D∘∑ h_{k}^{H}∘η_{∗}} & H^{i+1}(P H \ ⋃ P A^{H})}

Here α ∈ H^{2}(P H) is the canonical generator. Also note H_{∗}(ΔQ_{[0,n]}, ΔQ_{[0,n)}) = 0, since Q_{[0,n]} has a minimal element, which explains the missing of a summand in the first row. The map π: A \ ⋃ A^{A} → P H \ ⋃ P A^{H} is a fibre bundle with fibre ℂ \ {0}, α is its Thom class, and the right column of the diagram is its Gysin sequence.

If we start with a linear arrangement A in V , pass to the arrangement A × ℂ in V × ℂ, and set H := V × {0}, then P (A × ℂ)^{H} will be P A and (A × ℂ)^{A} will be isomorphic to the original arrangement A. The map π constructed above will correspond to the quotient map V \ ⋃ A → P V \ ⋃ P A, so that we have just described its cohomology Gysin sequence completely combinatorially. Just the isomorphism in the second row is possibly not as explicit as we would like it. This, however, will be remedied in a minute, when we study the intersection with the hyperplane H to obtain i_{!} ∘ h_{k} = h_{k−1}^{H} ∘ η_{∗}.

Intersecting with a hyperplane

We will describe the map on the homology of an affine or a projective arrangement given by intersecting with a hyperplane, that is the transfer map of the inclusion of the hyperplane. In contrast to the material covered so far in this chapter, with the exception of the results mentioning cohomology, this depends on the projective or affine space in which the arrangement is contained being a manifold. The same is of course true for the intersection products treated later on, and the results obtained here will be the basis for the inductive step in the proof of the product formulas.

Let Λ^{H}: V → 𝕂 be a linear functional that vanishes on no element of Q_{[0,2n]} and H := ker Λ^{H}. A induces an arrangement A^{H} := {A ∩ H: A ∈ A} in H. If we denote the intersection poset of A^{H} by Q^{H},

η: Q_{(0,n]} → Q_{[0,n−1]}^{H}
q ↦ q ∩ H

is an isomorphism lowering dimensions by one.

We consider the inclusion map i: (P H, ⋃ P A^{H}) → (P V, ⋃ P A).

2.1.35 Proposition. Let 𝕂 = ℂ. For c ∈ H_{∗}(ΔQ_{[k,n]}, ΔQ_{[k,n)}) we have

i_{!}(h_{k}(c)) = \begin{cases}h_{k−1}^{H}(η_{∗}(c)), & k > 0, \\ 0, & k = 0 \end{cases}

and in particular ker i_{!} = im h_{0}.

Proof. We first choose (x_{j}^{u})_{0≤j<k} with x_{j}^{u} ∈ u ∩ H satisfying the conditions of Definition 2.1.25 and therefore defining functions f_{H}^{k−1} and h_{k−1}^{H}. Now for each u ∈ Q_{[k,n]} we choose x_{k}^{u} ∈ u with Λ^{H}(x_{k}^{u}) = 1. (x_{j}^{u})_{0≤j≤k} then also satisfies the conditions of Definition 2.1.25 and can be used to define f^{k} and h_{k}.

Indeed we calculate

Λ^{H} (∑_{j=0}^{k} ∑_{i=0}^{r} μ_{j}λ_{i}x_{j}^{u_{i}} ) = ∑_{i=0}^{r} μ_{k}λ_{i} = μ_{k}.

First this implies that f^{k}(x, y) ∈ H iff x ∈ ℂP^{k−1} ⊂ ℂP^{k}. In particular f^{0} misses H, which proves that part of the proposition, and we now assume k > 0. The equation also implies that for x = [x_{0} : ⋯: x_{k−1}] ∈ ℂP^{k−1} and y ∈ Δ(Q_{[k,n]}) the map μ ↦ f^{k}([x_{0} : ⋯: x_{k−1} : μ], y) meets H transversally. Furthermore

f^{k} ([μ_{0} : ⋯: μ_{k−1} : 0], ∑_{i=0}^{r} λ_{i}u_{i}) = [∑_{j=0}^{k−1} ∑_{i=0}^{r} μ_{j}λ_{i}x_{j}^{u_{i}} ]

= f_{H}^{k−1} ([μ_{0} : ⋯: μ_{k−1}], ∑_{i=0}^{r} λ_{i}η(u_{i})) ,

which proves the proposition as we will now show in more detail.

Let ϑ ∈ H^{2}(P V, P V \ P H) be the Thom class of (the normal bundle of) P H in P V , i.e. the class satisfying θ ⌢ [P V] = [P H]. By the above calculations (f^{k})^{∗}(ϑ) ∈ H^{2} (ℂP^{k} × ΔQ_{[k,n]}, (ℂP^{k} \ ℂP^{k−1}) × ΔQ_{[k,n]}) is the Thom class of ℂP^{k−1} × ΔQ_{[k,n]} in ℂP^{k} × ΔQ_{[k,n]} which equals the class α × 1 where α ∈ H^{2}(ℂP^{k}, ℂP^{k} \ ℂP^{k−1}) which is again a Thom class and maps to the canonical generator of H^{∗}(ℂP^{k}). We finally calculate

i_{!}(h_{k}(c)) = ϑ ⌢ h_{k}(c)
= ϑ ⌢ f_{∗}^{k}([ℂP^{k}] × c)
= f_{∗}^{k} ((f^{k})^{∗}(ϑ) ⌢ ([ℂP^{k}] × c))
= f_{∗}^{k} ((α × 1) ⌢ ([ℂP^{k}] × c))
= f_{∗}^{k} ((α ⌢ [ℂP^{k}]) × (1 ⌢ c))
= f_{∗}^{k} ([ℂP^{k−1}] × c)
= h_{k−1}(η_{∗}(c))

as claimed. □

Turning to affine arrangements we let I ⊂ V be another hyperplane and consider the inclusion map i: (P H, P (I ∩ H) ∪ ⋃ P A^{H}) → (P V, P I ∪ ⋃ P A).

2.1.36 Proposition. Let A be a complex arrangement or a real ≥ 2-arrangement. Let I ⊂ V be a hyperplane (at infinity) and H ⊂ V a hyperplane in general position with respect to A ∪ {I}. Let u ∈ Q̄, c ∈ H_{∗}(Δ[u, V], Δ[u, V) ∪ Δ(u, V]) and

h_{u}: H_{∗}(Δ[u, V], Δ[u, V) ∪ Δ(u, V]) → H_{∗} (P V, P I ∪ ⋃ P A)

be the homomorphism from Definition 2.1.29. In case d(u) > 0, we also consider the homomorphism

h_{η(u)}^{H}: H_{∗}(Δ[η(u), η(V)], Δ[η(u), η(V)) ∪ Δ(η(u), η(V)])

→ H_{∗} (P H, (P H ∩ P I) ∪ ⋃ P A^{H}) .

In case 𝕂 = ℝ, we assume the orientations of (P u, P I ∩P u) and (P (u∩H), P (u∩ H ∩ I)) which are used in the definitions of h_{u} and h_{η(u)}^{H} to be related by

(f̄_{H}^{η(u)})_{∗}([ℝP^{d(u)−1}, ℝP^{d(u)−2}]) = i_{!} (f̄_{∗}^{u}([ℝP^{d(u)}, ℝP^{d(u)−1}])) ,  (2.43)

see (2.32).

Considering the inclusion map i: (P H, P (I ∩ H) ∪ ⋃ P A^{H}) → (P V, P I ∪ ⋃ P A) we have

i_{!}(h_{u}(c)) = \begin{cases}h_{η(u)}^{H}(η_{∗}(c)), & d(u) > 0, \\ 0, & d(u) = 0. \end{cases}

In particular ker i_{!} = ⨁_{u∈Q̄_{{0}}} h_{u}.

Proof. Let Λ^{I}: V → 𝕂 be a linear functional with ker Λ^{I} = I.

For d(u) = 0 we may assume that f^{u} and hence h_{u} is defined via points satisfying Λ^{H}(x_{v}^{u}) = 1 for all v ≥ u. Then

Λ^{H} ( ∑_{i=0}^{d(u)−1} μ_{i}b_{i}^{u} + μ_{d(u)} ∑_{j=0}^{r} λ_{j}x_{v_{j}}^{u} ) = Λ^{H} (∑_{j=0}^{r} λ_{j}x_{v_{j}}^{u} ) = 1  (2.44)

and f^{u} misses H so that i_{!}(h_{u}(c)) = 0.

Now let d(u) > 0. We may assume that f^{u} and hence h_{u} is defined via points satisfying b_{0}^{u}, …, b_{d(u)−2}^{u} ∈ u∩H ∩I, b_{d(u)−1}^{u} ∈ u∩I, Λ^{H}(b_{d(u)−1}^{u}) = 1, x_{u}^{u} ∈ u∩H, Λ^{I}(x_{u}^{u}) = 1, x_{v}^{u} ∈ v ∩ H ∩ I for v > u. For these

Λ^{H} ( ∑_{i=0}^{d(u)−1} μ_{i}b_{i}^{u} + μ_{d(u)} ∑_{j=0}^{r} λ_{j}x_{v_{j}}^{u} ) = μ_{d(u)−1}.  (2.45)

Hence, for y ∈ Δ[u, V] and (μ_{0}, …, μ_{d(u)−2}, μ_{d(u)}) ∈ 𝕂^{d(u)} \ {0} the map μ ↦ f^{u} ([μ_{0} : ⋯: μ_{d(u)−2} : μ : μ_{d(u)}], y) meets H transversally in 0 ∈ 𝕂. Furthermore

Λ^{I} ( ∑_{i=0}^{d(u)−1} μ_{i}b_{i}^{u} + μ_{d(u)} ∑_{j=0}^{r} λ_{j}x_{v_{j}}^{u} ) = \begin{cases}0, & v_{0} > u, \\ μ_{d(u)}λ_{0}, & v_{0} = u. \end{cases}  (2.46)

So f^{u}(x, y) ∈ I, iff x ∈ 𝕂P^{d(u)−1} or y ∈ Δ(u, V]. We set m := dim_{ℝ} 𝕂 and define ϑ ∈ H^{m}(P V \P I, P V \(P I ∪P H)) to be the Thom class of P H \P I in P V \P I, satisfying ϑ ⌢ [P V, P I] = [P H, P I ∩ P H]. It follows that (f^{u})^{∗}(ϑ) = εα × 1 with

ε ∈ {+1, −1} ,
α ∈ H^{m} (𝕂P^{d(u)} \ 𝕂P^{d(u)−1}, 𝕂P^{d(u)} \ (𝕂P^{d(u)−1} ∪ j[𝕂P^{d(u)−1}])) ,
1 ∈ H^{0}((Δ[u, V] \ Δ(u, V))),

where α satisfies α ⌢ [𝕂P^{d(u)}, 𝕂P^{d(u)−1}] = [j[𝕂P^{d(u)−1}], j[𝕂P^{d(u)−2}]].

Ignoring b_{d(u)−1}^{u}, these points also define a map f_{H}^{η(u)} : 𝕂P^{d(u)−1}×Δ[η(u), η(V)] → P H that induces h_{u}^{H} with

f^{u} ([μ_{0} : ⋯: μ_{d(u)−2} : 0 : μ_{d(u)}], ∑_{i=0}^{r} λ_{i}u_{i}) =

= [ ∑_{i=0}^{d(u)−2} μ_{i}b_{i}^{u} + μ_{d(u)} ∑_{j=0}^{r} λ_{j}x_{v_{j}}^{u} ] =

= f_{H}^{η(u)} ([μ_{0} : ⋯: μ_{d(u)−2} : μ_{d(u)}], ∑_{i=0}^{r} λ_{i}u_{i}) , (2.47)

i.e. f^{u}(j(x), y) = f_{H}^{η(u)}(x, η(y)). We can now calculate

i_{!}(h_{u}(c)) = ϑ ⌢ h_{u}(c)
= ϑ ⌢ f_{∗}^{u}([𝕂P^{d(u)}, 𝕂P^{d(u)−1}] × c)
= f_{∗}^{u} ((f^{u})^{∗}(ϑ) ⌢ ([𝕂P^{d(u)}, 𝕂P^{d(u)−1}] × c))
= f_{∗}^{u} ((εα × 1) ⌢ ([𝕂P^{d(u)}, 𝕂P^{d(u)−1}] × c))
= εf_{∗}^{u} ([j[𝕂P^{d(u)−1}], j[𝕂P^{d(u)−2}]] × c)
= ε(f_{H}^{η(u)})_{∗}([𝕂P^{d(u)−1}, 𝕂P^{d(u)−2}] × η_{∗}(c))
= εh_{η(u)}^{H}(η_{∗}(c)).

Similarly we find i_{!} (f̄_{∗}^{u}([ℝP^{d(u)}, ℝP^{d(u)−1}])) = ε(f̄_{H}^{η(u)})_{∗}([ℝP^{d(u)−1}, ℝP^{d(u)−2}]) and hence ε = 1 by comparison with (2.43) □

2.1.37 Remark. For affine ≥ 2-arrangements, Proposition 2.1.36 already almost solves the problem of describing the intersection product in combinatorial terms, as we will sketch now.

If A is an arrangement in X and B an arrangement in Y then the product (X, ⋃ A) × (Y, ⋃ B) equals (X × Y, ⋃(A × B)), where A × B is the arrangement in X × Y defined as {A × Y : A ∈ A} ∪ {X × B: B ∈ B}. Combinatorial descriptions of the homology of (X, ⋃ A) and of (Y, ⋃ B) easily lead to combinatorial descriptions of the homology of the product and the cross product map. For projective arrangements we will carry this out partially in Proposition 2.3.2. For linear arrangements, see [dLS01, Prop. 4.1].

Having obtained a description of the cross product, to describe intersection products it then remains to describe the intersection with the diagonal, i.e. the transfer map of the diagonal map. The product of two affine arrangements is again an affine arrangement, hence the diagonal is an affine plane itself, and this can be done by applying Proposition 2.1.36 n times. The only hindrance would be that the diagonal is not in general position with respect to the product arrangement, since the original arrangement is not in general position with respect to itself. This problem could be dealt with using methods similar to those we will apply in the proof of Proposition 2.3.13 on page 67.

However, the product of two projective spaces is not again a projective space. Therefore, the description of intersection products in projective arrangements requires additional techniques.

2.2 Products

Statement of results

We remind the reader of the product ˆ× from Definition 1.3.14.

2.2.1 Theorem. Let A be a complex arrangement. For all k, l ≥ 0 and all c ∈ H_{∗}(ΔQ_{[k,n]}, ΔQ_{[k,n)}), d ∈ H_{∗}(ΔQ_{[l,n]}, ΔQ_{[l,n)}) we have

h_{k}(c) • h_{l}(d) = \begin{cases}h_{k+l−n}(c ˆ× d), & k + l ≥ n, \\ 0, & k + l < n. \end{cases}  (2.48)

This is the main result of this work. Its proof will take up Section 2.3 and be completed on p. 68.

The statement of the corresponding result for affine arrangements will require some preparations, because we do not restrict ourselves to complex arrangements in this case.

The definition of the homomorphisms h_{u} in Definition 2.1.29 depends on the choice of a basis (b_{i}^{u})_{i} for u ∈ Q̄. This also orients all of the u. In the case of a ≥ 2-arrangement this orientation determines h_{u}. Orientations for u and v with u + v = V determine (together with the orientation of V ) an orientation of u ∩ v. Depending on whether this agrees with the orientation of u ∩ v defined by (b_{i}^{u∩v})_{i} or not, we set ε_{u,v} = 1 or ε_{u,v} = −1. For complex arrangements, every u ∈ Q̄ has a canonical orientation, and all of the ε_{u,v} will equal 1. We define these numbers more formally in the form in which we will use them.

2.2.2 Definition. Let A be a real arrangement and functions f^{u} for all u ∈ Q̄ chosen as in Definition 2.1.29. Let f̄^{u}be the function defined in (2.32). For u, v ∈ Q̄ with u + v = V and u ∩ v ⊄ H we define ε_{u,v} ∈ {+1, −1} by

f̄_{∗}^{u}([ℝP^{d(u)}, ℝP^{d(u)−1}]) • f̄_{∗}^{v}([ℝP^{d(v)}, ℝP^{d(v)−1}]) =

= ε_{u,v}f̄_{∗}^{u∩v}([ℝP^{d(u∩v)}, ℝP^{d(u∩v)−1}]), (2.49) where the intersection product is defined by dualizing

H^{∗}(P V \ P H, P V \ (P H ∪ P u)) ⊗ H^{∗}(P V \ P H, P V \ (P H ∪ P v))

⟶^{⌣} H^{∗}(P V \ P H, P V \ (P H ∪ P (u ∩ v))).

2.2.3 Theorem. Let A be a real ≥ 2–arrangement. For all u, v ∈ Q̄ and all c ∈ H_{r}(Δ[u, V], Δ[u, V) ∪ Δ(u, V]), d ∈ H_{s}(Δ[v, V], Δ[v, V) ∪ Δ(v, V]) we have

h_{u}(c) • h_{v}(d) = \begin{cases}(−1)^{r(n−d(u))}ε_{u,v}h_{u∧v}(c ˆ× d), \\ if u ∧ v ∈ Q̄ \\ and d(u) + d(v) = d(u ∧ v) + n, \\ 0, & otherwise. \end{cases}  (2.50)

The proof of this theorem will be completed at the end of this section, p. 57. This theorem has first been proved in [dLS01] (for central linear arrangements only) and in [DGM00], where it is stated in a somewhat different form.

2.2.4 Remark. If a complex arrangement is regarded as a real arrangement of double dimension, the bases can be chosen derived from complex bases, in which case all of the ε_{u,v} still equal 1. For a general real arrangement this cannot always be achieved. The dependence of the cup product in the cohomology of a linear arrangement on the numbers ε_{u,v} has first been shown in [Zie93], where it is used to construct two real arrangements with equalling intersection posets and dimension functions, one of them being a complex arrangement regarded as a real arrangement, with non-isomorphic cohomology rings.

2.2.5 Remark. Continuing Remark 2.1.32, the complex case of Theorem 2.2.3 can be seen as a special case of Theorem 2.2.1, since all of the isomorphisms in that remark are induced by inclusions and therefore respect the product ˆ×.

Graded formulas

We first prove graded versions of the theorems stated in the preceding section. These follow more or less for free from the algebraic machinery set up in Chapter 1. They state that the equations from the theorems we aim to prove hold at least up to error terms in higher degrees of the direct sum decompositions of the homology groups of the arrangements. They are therefore typical for the kind of result obtainable by spectral sequence arguments.

The graded versions will be an import part of the proofs of the exact versions. They hold in greater generality and in particular are true for arbitrary real arrangements, for which the exact versions fail. Our proof of the exact version for complex projective arrangements will need further more geometric arguments.

2.2.6 Proposition. Let 𝕂 = ℂ or let 𝕂 = ℝ and coefficients be in ℤ_{2}. For all k + l ≥ n and c ∈ H_{∗}(ΔQ_{[k,n]}, ΔQ_{[k,n)}), d ∈ H_{∗}(ΔQ_{[l,n]}, ΔQ_{[l,n)}) we have

h_{k}(c) • h_{l}(d) − h_{k+l−n}(c ˆ× d) ∈ ⨁_{i>k+l−n} im h_{i}.  (2.51)

Proof. We want to apply Proposition 1.3.20. As the map ζ there we take the map H(P (F^{Q}))⊗_{Q} B(Q) → S(P (D^{A})) obtained as the composition of (2.19) and (2.14). We set m := dim_{ℝ} 𝕂. The isomorphism ⨁_{k} H_{∗}(ΔQ_{[k,n]}, ΔQ_{[k,n)})[−mk] ≅ H(H(P (D^{A}))⊗_{Q} B(Q)⊗_{Q} K^{p}) obtained by composing (2.28) with the first arrow from (1.8) is given explicitly by

\begin{matrix}C_{r}(ΔQ_{[k,n]}, ΔQ_{[k,n)}) & → H_{mk}(P (D^{A})) ⊗_{Q} B(Q)_{r} ⊗_{Q} K^{p} \\ ⟨q_{0}, …, q_{r}⟩ & ↦ e_{k}^{q_{0}} ⊗ (q_{0} ← q_{0} ← ⋯ ← q_{r} ← ⊤) ⊗ 1 \end{matrix}

with 1 ∈ K^{p}(⊤) = R and e_{k}^{q} ∈ H_{mk}(P q) the canonical generator. By Proposition 2.1.25 the map h_{k} agrees with the composition of this isomorphism and the map ϕ_{mk} from Proposition 1.3.20. We note that the dimension n there equals mn in our current notation. To see that the product in that proposition agrees with ˆ× under the above isomorphism we only have to note that e_{k}^{p} • e_{l}^{q} = e_{k+l−n}^{p∧q} for k + l ≥ n. □

2.2.7 Proposition. Let u, v ∈ Q̄, c ∈ H_{∗}(Δ[u, V], Δ[u, V) ∪ Δ(u, V]), d ∈ H_{∗}(Δ[v, V], Δ[v, V) ∪ Δ(v, V]). If u ∧ v ∈ Q̄, then

h_{u}(c) • h_{v}(d) ∈ ⨁_{w≥u∧v} im h_{w}.  (2.52)

If additionally d(u) + d(v) = d(u ∧ v) + n, then

h_{u}(c) • h_{v}(d) − (−1)^{|c|(n−d(u))}ε_{u,v}h_{u∧v}(c ˆ× d) ∈ ⨁_{w>u∧v} im h_{w}  (2.53)

for 𝕂 = ℝ or

h_{u}(c) • h_{v}(d) − h_{u∧v}(c ˆ× d) ∈ ⨁_{w>u∧v} im h_{w}  (2.54)

for 𝕂 = ℂ. If u ∩ v ⊂ I, then

h_{u}(c) • h_{v}(d) ∈ ⨁_{w∈Q̄_{(d(u)+d(v)−n,n]}} im h_{w}.  (2.55)

Proof. We treat the real case only, as the easier case of 𝕂 = ℂ can be proved in the same way or derived from the case 𝕂 = ℝ.

Considering the arrangement {q ∈ Q: q ≥ u or q ≥ v}, the intersection poset of which can be considered as a subset of the interval [u ∧ v, V] in Q, we obtain (2.52) by naturality of the intersection product with respect to inlusion maps.

The rest of the proof proceeds as the preceding one, using a relative version of Proposition 1.3.20. We take up the notation from the proof of Proposition 2.1.29. The isomorphism

⨁_{u∈Q̄} H_{∗}(Δ[u, V], Δ[u, V) ∪ Δ(u, V])[−d(u)] ≅ H(H(A(D^{A})) ⊗_{Q} B(Q) ⊗_{Q} K^{p})

is induced by

\begin{matrix}C_{r}(Δ[u, V], Δ[u, V) ∪ Δ(u, V]) & → H_{d(u)}(A(D^{A})) ⊗_{Q} B(Q)_{r} ⊗_{Q} K^{p} \\ ⟨q_{0}, …, q_{r}⟩ & ↦ e^{u} ⊗ (u ← q_{0} ← … ← q_{r} ← V) ⊗ 1. \end{matrix}

with e^{u} := f̄_{∗}^{u}([ℝP^{d(u)}, ℝP^{d(u)−1}]) ∈ H_{d(u)}(P u, P (u ∩ I)), u ∈ Q̄. Under this isomorphism the map ϕ_{k} from Proposition 1.3.20 corresponds to ∑_{u∈Q̄_{{k}}} h_{u}.

For d(u) + d(v) = d(u ∧ v) + n we have e^{u} • e^{v} = ε_{u,v}e^{u∧v} by definition of ε_{u,v} and hence

h_{u}(c) • h_{v}(d) − (−1)^{|c|(n−d(u))}ε_{u,v}h_{u∧v}(c ˆ× d) ∈ ⨁_{w∈Q̄_{(k+l−n,n]}} im h_{w}.

Since

⨁_{w∈Q̄_{(k+l−n,n]}} im h_{w} ∩ ⨁_{w≥u∧v} im h_{w} = ⨁_{w>u∧v} im h_{w},

this proves (2.53).

For u ∩ v ⊂ I, e^{u} • e^{v} = 0 by necessity, since H(A(u ∧ v)) = 0, and the above argument yields (2.55). □

2.2.8 Remark. Proposition 2.2.7 is proved in [dLS01] as Theorem 7.5. While there it is more of an afterthought to the exact version, it appears here as a very natural result in its own right and a possible basis to a proof of the exact version.

2.2.9 Remark. If n is odd, then ℝP^{n} is orientable and Proposition 1.3.20 is applicable to H_{∗}(P V, ⋃ P A) also for 𝕂 = ℝ, R = ℤ, since in the proof of Proposition 2.1.20 the needed ZŽ–map is constructed. In the product H(P u)⊗H(P v) ⟶^{•} H(P (u∧v)) the product of two generators is a generator whenever possible and to determine the sign of the product of two generators of ℤ-summands, orientation information is needed as in the affine case.

Inductive proofs of product formulas

We now reduce Theorem 2.2.1 and Theorem 2.2.3 to the cases where they state that products be zero because of their degrees. This is done using the graded versions presented in the preceding section. The inductive step is made possible by the results on intersections with a hyperplane presented in Section 2.1.

2.2.10 Proposition. If it is true for all complex arrangements A of all dimensions that h_{k}(c)•h_{l}(d) = 0 for c ∈ H_{∗}(ΔQ_{[k,n]}, ΔQ_{[k,n)}), d ∈ H_{∗}(ΔQ_{[l,n]}, ΔQ_{[l,n)}), whenever k + l < n, then Theorem 2.2.1 is true.

Proof. We have to show that h_{k}(c) • h_{l}(d) = h_{k+l−n}(c ˆ× d), where we use the convention that h_{i} = 0 for i < 0.

We proceed by induction on the dimension of V .

For k + l < n the conjecture is covered by the assumption. For k + l ≥ n and dim V = 1 it is covered by Proposition 2.2.6 (and trivial anyway). For k + l ≥ n and dim V > 1 we choose a hyperplane in general position with respect to the arrangement and adopt the notation of Section 2.1. By induction and Proposition 2.1.35 (for k + l > n) or the assumption (for k + l = n)

i_{!}(h_{k}(c) • h_{l}(d)) = i_{!}(h_{k}(c)) • i_{!}(h_{l}(d)) = h_{k−1}^{H}(η_{∗}(c)) • h_{l−1}^{H}(η_{∗}(d)) =

= h_{k+l−n−1}^{H}(η_{∗}(c) ˆ× η_{∗}(d)) = h_{k+l−n−1}^{H}(η_{∗}(c ˆ× d)) = i_{!}(h_{k+l−n}(c ˆ× d)).

Again by Proposition 2.1.35, this implies h_{k}(c) • h_{l}(d) − h_{k+l−n}(c ˆ× d) ∈ ker i_{!} = im h_{0}. But h_{k}(c) • h_{l}(d) − h_{k+l−n}(c ˆ× d) ∈ ⨁_{i>k+l−n} im h_{i} by Proposition 2.2.6. Therefore h_{k}(c) • h_{l}(d) − h_{k+l−n}(c ˆ× d) = 0. □

2.2.11 Proposition. If it is true for all real ≥ 2-arrangements A of all dimension and all u, v ∈ Q̄ that h_{u}(c)•h_{v}(d) = 0 for c ∈ H_{∗}(Δ[u, V], Δ[u, V)∪Δ(u, V]) and d ∈ H_{∗}(Δ[v, V], Δ[v, V) ∪ Δ(v, V]) whenever d(u) + d(v) < n, then Theorem 2.2.3 is true.

Proof. The proof proceeds by induction on the dimension of V , parallel to the preceding one.

Let u, v ∈ Q̄, d(u) + d(v) ≥ n.

If dim V = 1, then Theorem 2.2.3 holds trivially for V . We assume dim V > 1 and let H ⊂ V be a hyperplane in general position with respect to A and I. For simplicity, we also assume d(u), d(v) > 0, because otherwise u = V or v = V and these cases are easily dealt with directly.

If u ∩ v ⊂ I, then η(u) ∩ η(v) ⊂ I ∩ H. If u ∧ v ∈ Q̄ and d(u) + d(v) − d(u ∧ v) < n then η(u) ∩ η(v) ⊂ I ∩ H or d(η(u)) + d(η(v)) − d(η(u) ∧ η(v)) < n − 1. In these cases

i_{!}(h_{u}(c) • h_{v}(d)) = i_{!}(h_{u}(c)) • i_{!}(h_{v}(d)) = h_{η(u)}^{H}(η_{∗}(c)) • h_{η(v)}^{H}(η_{∗}(d)) = 0

and hence h_{u}(c) • h_{v}(d) ∈ ⨁_{w∈Q̄_{{0}}} h_{w} by induction and Proposition 2.1.36, and h_{u}(c) • h_{v}(d) ∈ ⨁_{w∈Q̄_{(0,n]}} h_{w} by Proposition 2.2.7. Therefore h_{u}(c) • h_{v}(d) = 0.

Now let u + v = V , u ∩ v ⊄ I. If d(u) + d(v) = n, then

i_{!}(h_{u}(c) • h_{v}(d) − (−1)^{|c|(n−d(u))}ε_{u,v}h_{u∧v}(c ˆ× d)) =

= h_{η(u)}^{H}(η_{∗}(c)) • h_{η(v)}^{H}(η_{∗}(d)) = 0

by induction and Proposition 2.1.36. If d(u) + d(v) > n, we make sure that h_{η(u)}^{H}, h_{η(v)}^{H}, h_{η(u∧v)}^{H} are defined with orientations compatible with those underlying h_{u}, h_{v}, h_{u∧v} as in Proposition 2.1.36. Then

(f̄_{H}^{η(u)})_{∗}([ℝP^{d(η(u))}, ℝP^{d(η(u))−1}]) • (f̄_{H}^{η(v)})_{∗}([ℝP^{d(η(v))}, ℝP^{d(η(v))−1}]
= i_{!}(f̄_{∗}^{u}([ℝP^{d(u)}, ℝP^{d(u)−1}]) • f̄_{∗}^{v}([ℝP^{d(v)}, ℝP^{d(v)−1}])
= i_{!}(ε_{u,v}(f̄_{H}^{u∧v})_{∗}([ℝP^{d(u∧v)}, ℝP^{d(u∧v)−1}])
= ε_{u,v}(f̄_{H}^{η(u)∧η(v)})_{∗}([ℝP^{d(η(u)∧η(v))}, ℝP^{d(η(u)∧η(v))−1}])

and hence ε_{η(u),η(v)}^{H} = ε_{u,v}. Again, this yields

i_{!}(h_{u}(c) • h_{v}(d) − (−1)^{|c|(n−d(u))}ε_{u,v}h_{u∧v}(c ˆ× d)) =

h_{η(u)}^{H}(η_{∗}(c)) • h_{η(v)}^{H}(η_{∗}(d)) − (−1)^{|c|(n−1−d(η(u)))}ε_{η(u),η(v)}^{H}h_{η(u)∧η(v)}^{H}(c ˆ× d)

= 0.

In both cases, we combine this with Proposition 2.1.36 and Proposition 2.2.7 to get h_{u}(c) • h_{v}(d) − (−1)^{|c|(n−d(u))}ε_{u,v}h_{u∧v}(c ˆ× d) ∈ ⨁_{w∈Q̄_{{0}}} h_{w} ∩ ⨁_{w>u∧v} h_{w} = 0. □

Vanishing for affine arrangements

For affine ≥ 2-arrangements the vanishing of intersection products in the cases required by Proposition 2.2.11 is easily proved, since it is indeed possible to find chains representing the involved hology classes that do not intersect geometrically.

2.2.12 Proposition. Let A be a ≥ 2–arrangement, u, v ∈ Q̄, d(u) + d(v) < n, c ∈ H_{∗}(Δ[u, V], Δ[u, V) ∪ Δ(u, V]), d ∈ H_{∗}(Δ[u, V], Δ[u, V) ∪ Δ(u, V]). Then h_{u}(c) • h_{v}(d) = 0.

Proof. Let Λ be a linear functional on V with ker Λ = H.

There is a linear functional F ≠ 0 on V with u ⊂ ker F and v ∩ H ⊂ ker F. We choose b^{u}, x^{u} to define f^{u} and b̃^{v}, x̃^{v} to define f̃^{v} as in Definition 2.1.29. This can be done in such a way that F (x_{w}^{u}) ≤ 0 for all w ≥ u, F (x_{w}^{v}) ≥ 0 for all w ≥ v, F (x_{V}^{u}) < 0, F (x_{V}^{v}) > 0 and Λ(x_{u}^{u}) = Λ(x̃_{v}^{v}) = 1.

We set

X^{+} := {[z] ∈ P V : Λ(z) ≠ 0, Λ(z)^{−1}F (z) ≥ 0} ∪ P H,
X^{−} := {[z] ∈ P V : Λ(z) ≠ 0, Λ(z)^{−1}F (z) ≤ 0} ∪ P H.

Then

im f^{u} ⊂ X^{−}, (f^{u})^{−1}[X^{+} ∩ X^{−}] ⊂ 𝕂P^{k} × Δ[u, V) ∪ 𝕂P^{k−1} × Δ[u, V],
im f̃^{v} ⊂ X^{+}, (f̃^{v})^{−1}[X^{+} ∩ X^{−}] ⊂ 𝕂P^{l} × Δ[v, V) ∪ 𝕂P^{l−1} × Δ[v, V].

Therefore im f^{u} ∩ im f̃^{v} ⊂ ⋃ P A ∪ P H and h_{u}(c) • h_{v}(d) = 0. □

Proof of Theorem 2.2.3. The theorem follows directly from Proposition 2.2.12 and Proposition 2.2.11. □

2.3 Products in projective arrangements

In this section we will complete the proof of Theorem 2.2.1, the product formula for complex projective arrangements. We have already reduced this in Proposition 2.2.10 to the case of h_{k}(c) • h_{l}(d) with k + l < n, which we have to show to be zero. The corresponding fact for affine arrangements could be proved by representing the homology classes by chains which do not intersect geometrically. A proof along these lines seems not to be available for projective arrangements. We will first consider an example of a real projective arrangement where an intersection product of this kind is indeed not zero. Doing this we will also try to gain some intuition on why in the projective case the intersection of the chains should not make a homological contribution, even if existing geometrically. We will then develop the techniques necessary to transform this intuition into a proof.

An example of real projective arrangements

We will see how the product formula of Theorem 2.2.1 fails for real projective arrangements and sketch the difference between real and complex arrangements that will allow us to prove the formula for complex projective arrangements.

Let k, l ≥ 0, n := k + l + 1. We consider the following subspaces of ℝ^{n+1} = ℝ^{k} × ℝ^{l} × ℝ^{2}.

u := ℝ^{k} × {0} × (ℝ · (0, 1)),  ũ := {0} × ℝ^{l} × (ℝ · (1, 1)),
v := ℝ^{k} × {0} × (ℝ · (4, 1)),  ṽ := {0} × ℝ^{l} × (ℝ · (5, 1)),

In the arrangement Ǎ:= {u, v, ũ, ṽ} we will find classes c ∈ H(ΔQ_{[k,n]}, ΔQ_{[k,n)}), d ∈ H(ΔQ_{[l,n]}, ΔQ_{[l,n)}) with h_{k}(c) • h_{l}(d) ≠ 0, although k + l < n.

The combinatorial data of Ǎ are given by the intersection poset

[figure: V; Q̌_{[0,n]} =; u; v; ũ; ṽ; u ∩ v; ũ ∩ ṽ]

with u ∩ v and ũ ∩ ṽ only present for k > 0 and l > 0 respectively, and the dimensions d(u) = d(v) = k, d(u ∩ v) = k − 1, d(ũ) = d(ṽ) = l, d(ũ ∩ ṽ) = l − 1, d(V) = n. In case of k = l, we can, if we want to, avoid the intersections u∩v and ũ∩ṽ by a small change of u and ũ without substantially affecting the calculations below. This shows that, in contrast to the case of affine arrangements, a simple condition on the occuring codimensions will not be enough for the product formula to extend from complex to real arrangements.

To simplify the pictures below and to have the notation parallel that of Section 2.3, we consider Ǎ to be the union of the two arrangements A := {u, v} and Ã := {ũ, ṽ}. The arrangement Ã is in general position with respect to A as in Definition 2.3.4.

Denoting the intersection posets of A and Ã by Q and Q̃ respectively, we have H_{1}(ΔQ_{[k,n]}, ΔQ_{[k,n)}; ℤ_{2}) ≅ ℤ_{2}, generated by c := [⟨u, V ⟩ + ⟨v, V ⟩], and H_{1}(ΔQ̃_{[l,n]}, ΔQ̃_{[l,n)}; ℤ_{2}) ≅ ℤ_{2}, generated by d := [⟨ũ, V ⟩ + ⟨ṽ, V ⟩]. For the definition of

f^{k}: ℝP^{k} × (ΔQ_{[k,n]}, ΔQ_{[k,n)}) → (V, ⋃ P A),
f̃^{l}: ℝP^{l} × (ΔQ̃_{[k,n]}, ΔQ̃_{[k,n)}) → (V, ⋃ P Ã),

and hence of

h_{k}: H_{r}(ΔQ_{[k,n]}, ΔQ_{[k,n)}; ℤ_{2}) → H_{r+k}(V, ⋃ P A; ℤ_{2}),
h̃_{l}: H_{r}(ΔQ̃_{[k,n]}, ΔQ̃_{[k,n)}; ℤ_{2}) → H_{r+l}(V, ⋃ P Ã; ℤ_{2})

we set, with (e_{0}, …, e_{n}) the standard basis of V = ℝ^{n+1},

x_{j}^{u} := \begin{cases}e_{j}, & j < k, \\ e_{k+l+1}, & j = k, \end{cases}  x̃ũ ; j := \begin{cases}e_{k+j}, & j < l, \\ e_{k+l} + e_{k+l+1}, & j = l, \end{cases}

x_{j}^{v} := \begin{cases}e_{j}, & j < k, \\ 4e_{k+l} + e_{k+l+1}, & j = k, \end{cases}  x̃ṽ ; j := \begin{cases}e_{k+j}, & j < l, \\ 5e_{k+l} + e_{k+l+1}, & j = l, \end{cases}

x_{j}^{V} := \begin{cases}e_{j}, & j < k, \\ 2e_{k+l} + e_{k+l+1}, & j = k, \end{cases}  x̃_{j}^{V} := \begin{cases}e_{k+j}, & j < l, \\ 3e_{k+l} + e_{k+l+1}, & j = l. \end{cases}

To determine h_{k}(c) • h_{l}(d), we first have a look at the geometric intersection S := f^{k}[ℝP^{k} ×ΔQ_{[k,n]}]∩f^{l}[ℝP^{l} ×ΔQ̃_{[l,n]}]. For x ∈ Δ(Q_{[k,n]}), y ∈ Δ(Q̃_{[l,n]}), the intersection f [ℝP^{k} × {x}] ∩ f [ℝP^{l} × {y}] is either empty or consists of a single point. The left of the following two pictures shows the two dimensional simplicial complex ΔQ_{[k,n]} × ΔQ̃_{[l,n]} = Δ(Q_{[k,n]} × Q̃_{[l,n]}).

[figure: (u, ṽ); (V, ṽ); •; (v, ṽ); ṽ; (u, V); (V, V); (v, V); u; V; v; •; (u, ũ); (V, ũ); (v, ũ); ũ]  (2.56)

The dotted line depicts the set S̄ of those points (x, y) for which the intersection is nonempty. S is a connected 1–dimensional manifold with boundary, and we can see from the picture that one boundary point lies in u∩V = u and the other one in V ∩ṽ = ṽ. A closer look at S, which is the intersection of two manifolds that meet transversely, shows that indeed h_{k}(c)•h̃_{l}(d) = ȟ_{0}([⟨u, V ⟩+⟨ṽ, V ⟩]). [⟨u, V ⟩+⟨ṽ, V ⟩] is a generator of H_{1}(ΔQ̌_{[0,n]}, ΔQ̌_{[0,n)}; ℤ_{2}), therefore h_{k}(c) • h̃_{l}(d) ≠ 0.

We equip Q × Q̃ with a dimension function d(p, q) := d(p) + d(q). The map Q × Q̃ → Q̌, (p, q) ↦ p ∩ q, sends ((Q × Q̃)_{[n,2n]}, (Q × Q̃)_{[n,2n)}) to (Q̌_{[0,n]}, Q̌_{[0,n)}).

In the picture, the border of the square is Δ(Q_{[k,n]} × Q̃_{[l,n]})_{[0,2n)} and the four vertices at the corners are Δ(Q_{[k,n]} × Q̃_{[l,n]})_{[0,n)}. Under the composition of maps

(Δ(Q_{[k,n]} × Q̃_{[l,n]}) \ Δ(Q_{[k,n]} × Q̃_{[l,n]})_{[0,n)}, ; Δ(Q_{[k,n]} × Q̃_{[l,n]})_{[0,2n)} \ Δ(Q_{[k,n]} × Q̃_{[l,n]})_{[0,n)})

(Δ(Q_{[k,n]} × Q̃_{[l,n]})_{[n,2n]}, Δ(Q_{[k,n]} × Q̃_{[l,n]})_{[n,2n)})

(ΔQ̌_{[0,n]}, ΔQ̌_{[0,n)}),

(2.57)

the first being a deformation retraction and the second given by inclusion, in our example (S̄, 𝔡S̄) is mapped to the dotted line in the picture on the right. This set carries the relative cycle ⟨u, V ⟩ + ⟨ṽ, V ⟩ representing h_{k}(c) • h̃_{l}(d). We will see in Section 2.3 that this is not just a coincidence.

When considering complex arrangements we will see that in the above situation we gain one dimension compared to real arrangements, and S̄ will miss the cone with top the vertix (V, V) and base Δ(Q_{[k,n]} × Q̃_{[l,n]})_{[0,n)}. The map of (S̄, 𝔡S̄) to (ΔQ̌_{[0,n]}, ΔQ̌_{[0,n)}) will therefore miss the vertix V and be homotopic to a map with image in ΔQ̌_{[0,n)}. This is the idea behind the proof of Proposition 2.3.14, although it will be technically a bit different.

The product of two arrangements and general position

The intersection of two sets can be identified with the intersection of their cartesian product with a diagonal. Similarly the intersection product of two homology classes equals the image of their cross product under the image of the transfer map associated with the diagonal map. We therefore study the products of two arrangements.

Already in the real example we have discussed, it was useful to assume the homology classes of which the product was to be determined to be carried by different arrangements. So we now assume to be given a second arrangement Ã in V with intersection poset Q̃. For the first part of this section the arrangement could be in a vector space different from V , but we will have no use for this generality later on.

We equip the poset Q × Q̃ with a dimension function d by d(u, v) := d(u) + d(v). The counterpart of h for Ã will be denoted by h̃ and so on.

As noted above, we will be interested in cross products.

2.3.1 Definition and Proposition. Any choice of (y_{i}^{u,v})_{i=0,...,k}, (z_{i}^{u,v})_{i=0,...,l} for (u, v) ∈ Q_{[k,n]} × Q̃_{[l,n]} with y_{i}^{u,v} ∈ u, z_{i}^{u,v} ∈ v and such that for all (u_{0}, v_{0}) < ⋯ < (u_{m}, v_{m}) and λ ∈ Δ^{m} the system (∑_{j} λ_{j}y_{i}^{u_{j},v_{j}})_{i} as well as the system (∑_{j} λ_{j}z_{i}^{u_{j},v_{j}})_{i} is linearly independent, yields a map

g: ℂP^{k}×ℂP^{l}×(Δ(Q_{[k,n]}×Q̃_{[l,n]}),Δ(Q_{[k,n]}×Q̃_{[l,n]})_{[0,2n)})→(P V,⋃ P A)×(P V,⋃ P Ã)
([μ_{0}:···:μ_{k}],[ν_{0}:···:ν_{l}],∑_{j} λ_{j}(u_{j},v_{j}))↦([∑_{i,j} λ_{j}μ_{i}y_{i}^{uj ,vj} ],[∑_{i,j} λ_{j}ν_{i}z_{i}^{uj ,vj} ]).

As in Definition 2.1.25, any two such maps are homotopic for 𝕂 = ℂ. □

2.3.2 Proposition. Let c ∈ H_{∗}(ΔQ_{[k,n]}, ΔQ_{[k,n)}), d ∈ H_{∗}(ΔQ̃_{[l,n]}, ΔQ̃_{[l,n)}), and 𝕂 = ℂ. Then h_{k}(c) × h̃_{l}(d) = g_{∗} ([ℂP^{k}] × [ℂP^{l}] × (c × d)).

Proof. Since 𝕂 = ℂ, the homomorphisms h_{k}, h̃_{l} and g_{∗} do not depend on the choices made in defining them. For the choice y_{i}^{u,v} = x_{i}^{u}, z_{i}^{u,v} = x̃_{i}^{v}, we just get the map f^{k} × f̃^{l} up to identification of

ℂP^{k} × ℂP^{l} × (Δ(Q_{[k,n]} × Q̃_{[l,n]}), Δ(Q_{[k,n]} × Q̃_{[l,n]})_{[0,2n)})

with ℂP^{k} × (ΔQ_{[k,n]}, ΔQ_{[k,n)}) × ℂP^{l} × (ΔQ̃_{[l,n]}, ΔQ̃_{[l,n)}). Again since 𝕂 = ℂ, no sign is introduced by the interchange of factors made in this identification. □

As noted after discussing the real example, it will be important to control the codimension of a set corresponding to the dotted line in (2.56). We will now work towards this and start with an algebraic lemma.

2.3.3 Lemma. Let 𝕂 = ℂ. Let u, v be subspaces of V in general position with respect to each other, dim u = r ≥ k + 1, dim v = s ≥ l + 1, dim V = n + 1, k + l < n. Let O be the open subspace of the affine space u^{k+1} × v^{l+1} defined by

O := {(y_{0}, …, y_{k}, z_{0}, …, z_{l}): dim(span {y_{i}}) = k + 1, dim(span {z_{i}}) = l + 1}

and algebraic subsets ⋯ ⊂ S_{1} ⊂ S_{0} ⊂ O defined by

S_{m} := {(y_{0}, …, y_{k}, z_{0}, …, z_{l}): dim(span({y_{i}} ∪ {z_{j}})) < k + l + 2 − m}

Then S_{m} \ S_{m+1} is a complex submanifold of codimension (1 + m)(n − k − l + m).

Proof. We consider (y_{0}, …, y_{k}, z_{0}, …, z_{l}) ∈ S_{m} \ S_{m+1}. This implies u + v = V . We set Y := span {y_{i}}, t := n − k − s + dim(Y ∩ v), and choose a basis (e_{0}, …, e_{n}) of V such that span {e_{0}, …, e_{r−1}} = u, span {e_{n−s+1}, …, e_{n}} = v, span {e_{t}, …, e_{k+1}} = Y . Let A be the (n + 1) × (k + l + 2)–matrix with columns (y_{0}, …, y_{k}, z_{0}, …, z_{l}) expressed using this basis. Elements of O are represented by matrices A′ = (a′_{ij}) with a′_{ij} = 0 for r ≤ i ≤ n, 0 ≤ j ≤ k and for 0 ≤ i ≤ n − s, k+1 ≤ j ≤ k+l+1 such that the first k+1 and the last l+1 columns are linearly independent, and A = (a_{ij}) has the additional property that the first t rows are zero.

There are sets I and J with {t, …, k + t} ⊂ I ⊂ {t, …, n}, {0, …, k} ⊂ J ⊂ {0, …, k + l + 1} and |I| = |J| = k + l + 1 − m such that the matrix B := (a_{ij})_{i ∈ I,j ∈ J} is regular. Similarly, there exist I′, J′ with I′ ⊂ {t, …, n} \ M, {k + 1, …, k + l + 1} ⊂ J′ ⊂ {0, …, k + l + 1} and |I|′ = |J|′ = k + l + 1 − m such that the matrix C := (a_{ij})_{i ∈ I′,j ∈ J′} is regular.

Let U ⊂ O be a neighbourhood of A such that for every A′ = (a′_{ij}) ∈ U the matrices (a′_{ij})_{i ∈ I,j ∈ J} and (a′_{ij})_{i ∈ I′,j ∈ J′} are regular. Then an A′ ∈ U is in S_{m} if and only if the equations

f_{i_{0}j_{0}}(A′) := det(a′_{ij})\begin{matrix}i∈I∪\{i_{0}\} \\ j∈J∪\{j_{0}\} \end{matrix} = 0 for all i_{0} ∈ I_{0} := \{n + 1 − s, …, n\} \ I,
j_{0} ∈ J_{0} := {0, …, k + l + 1} \ J

and

g_{i_{0}j_{0}}(A′) := det(a′_{ij})\begin{matrix}i∈I′∪\{i_{0}\} \\ j∈J′∪\{j_{0}\} \end{matrix} = 0 for all i_{0} ∈ \{0, …, t − 1\} ,
j_{0} ∈ J′_{0} := {0, …, k + l + 1} \ J′

hold. To see this, assume A′ ∉ S_{m}, i.e. rk A′ > k + l + 1 − m. If the rank of the matrix A′ with the first t rows deleted is greater than k + l + 1 − m, one of the functions f_{i_{0}j_{0}} becomes non-zero, otherwise one of the functions g_{i_{0}j_{0}} .

Finally we compute for (i_{1}, j_{1}) ∈ I_{0} × J_{0}

|\frac{∂f_{i_{0}j_{0}}(A)}{∂a_{i_{1}j_{1}}} | = \begin{cases}|det B|, & (i_{0}, j_{0}) = (i_{1}, j_{1}), \\ 0, & (i_{0}, j_{0}) ≠ (i_{1}, j_{1}), \end{cases} (i_{0}, j_{0}) ∈ I_{0} × J_{0},
\frac{∂g_{i_{0}j_{0}}(A)}{∂a_{i_{1}j_{1}}} = 0,  (i_{0}, j_{0}) ∈ \{0, …, t − 1\} × J′_{0}

and for (i_{1}, j_{1}) ∈ {0, …, t − 1} × J′_{0}

\frac{∂f_{i_{0}j_{0}}(A)}{∂a_{i_{1}j_{1}}} = 0,  (i_{0}, j_{0}) ∈ I_{0} × J_{0},

|\frac{∂g_{i_{0}j_{0}}(A)}{∂a_{i_{1}j_{1}}} | = \begin{cases}|det C|, & (i_{0}, j_{0}) = (i_{1}, j_{1}), \\ 0, & (i_{0}, j_{0}) ≠ (i_{1}, j_{1}), \end{cases} (i_{0}, j_{0}) ∈ \{0, …, t − 1\} × J′_{0}

and |I_{0}×J_{0}∪{0, …, t − 1}×J′_{0}| = (n+1−|I|)(m+1) = (n−k−l+m)·(m+1). □

2.3.4 Definition. We say that the arrangement Ã is in general position with respect to the arrangement A, if for all u ∈ Q and v ∈ Q̃, we have u ∩ v = 0 whenever d(u) + d(v) < n and d(u ∩ v) = d(u) + d(v) − n otherwise.

2.3.5 Proposition. Let 𝕂 = ℂ, k + l < n, D ⊂ P V × P V be the diagonal and S ⊂ Δ(Q_{[k,n]} × Q_{[l,n]}) be defined as the set of all points x such that g [ℂP^{k} × ℂP^{l} × {x}] ∩ D ≠ Ø. For a generic choice of the points y_{i}^{u,v} and z_{i}^{u,v} defining g, the set S intersects every open simplex of Δ(Q_{[k,n]} × Q_{[l,n]}) in an algebraic set of real codimension 2(n − k − l).

Proof. In regard of Lemma 2.3.3 all that is required is that for each chain (u_{0}, v_{0}) < ⋯ < (u_{t}, v_{t}) the affine plane in u_{t}^{k+1} × v_{t}^{k+1} spanned by the t + 1 points (y^{u_{0},v_{0}}, z^{u_{0},v_{0}}), …, (y^{u_{t},v_{t}}, z^{u_{t},v_{t}}) meets the algebraic set S_{0} transversely. Assuming that the affine plane spanned by the first t of these points already meets S_{0} transversely, this will be fulfilled for a generic choice of (y^{u_{t},v_{t}}, z^{u_{t},v_{t}}) ∈ u_{t}^{k+1} × v_{t}^{k+1}. □

Recovering the direct sum decomposition

When discussing the real example, it seemed plausible that a certain subset of the order complex of the intersection poset should carry the inverse image of the considered intersection product under the isomorphism ∑_{k} h_{k}. We now develop tools that allow to actually prove this kind of proposition.

More generally, given a class in H_{∗} (P V, ⋃ P A) we want to determine the corresponding element of ⨁_{k} H_{∗}(ΔQ_{[k,n]}, ΔQ_{[k,n)}). Because of Proposition 2.1.35 it will suffice to identify the part in the summand H_{∗}(ΔQ_{[0,n]}, ΔQ_{[0,n)}). The key to this will be to not only consider the map f^{0} : ΔQ_{[0,n]} → P V , but also a map P V → Q_{[0,n]}, where the poset Q is topologized in an appropriate way yielding the space of strata. While we have up to this point used only the former map, in [DGM00] a description of the cohomology ring of the complement of an affine arrangement is obtained using exclusively the latter map. Here the interplay of both maps will be important.

2.3.6 Definition. Let P be a poset. We make P into a topological space by calling a set O ⊂ P open, iff x ∈ O implies y ∈ O for all y ≥ x.

2.3.7 Lemma. Let X be a space, P a poset, A ⊂ X, R ⊂ P. If f, g: (X, A) → (P, R) are continuous maps with f(x) ≥ g(x) for all x ∈ X, then f ≃ g.

Proof. The desired homotopy is given by

H: (X, A) × I → (P, R)
(x, t) ↦ \begin{cases}f(x), & t < 1, \\ g(x), & t = 1. \end{cases}

This map is continuous, since g^{−1}[O] ⊂ f^{−1}[O] for open O ⊂ X, and therefore H^{−1}[O] = f^{−1}[O] × [0, 1) ∪ g^{−1}[O] × {1} = f^{−1}[O] × [0, 1) ∪ g^{−1}[O] × I. □

2.3.8 Lemma. If P has a minimum or a maximum, then P is contractible.

Proof. By the preceding lemma, the constant map to the minimum respectively the maximum is homotopic to the identity. □

2.3.9 Definition. Let X be a space equipped with a covering ℭ by closed sets and let P be the poset P := {⋂ M: Ø ≠ M ⊂ ℭ, ⋂ M ≠ Ø}, odered by inclusion. We define a continous map

s: X → P
x ↦ min {p ∈ P: x ∈ P } = ⋂ {C ∈ ℭ: x ∈ C} .

In particular we consider the following two kinds of maps. For our arrangement A we consider the map s^{A}: P V → Q_{[0,n]} corresponding to the covering P A ∪ {P V } of P V . For a poset P which has unique minima in the sense that for M ⊂ P, M ≠ Ø, the set {p ∈ P: p ≤ q for all q ∈ M } is either empty or of the form {p: p ≤ q} for a q ∈ P, we consider the map s^{P} : ΔP → P arising from the covering of ΔP by the subspaces Δ({p′: p′ ≤ p}), p ∈ P.

2.3.10 Lemma. For a finite poset P and R ⊂ P, both satisfying the condition regarding minima of the preceding definition, the map s_{∗}^{P} : H_{∗}(ΔP, ΔR) ⟶^{≅} H_{∗}(P, R) is an isomorphism.

Proof. We may assume R = Ø, because the general case will follow by an application of the five lemma.

We consider the covering of P by the open subsets X(p) := {q: q ≥ p}. These together with the inclusion maps form a P -diagram of spaces. If ℤ denotes the constant diagram, then S(X)⊗_{P^{o}}ℤ = ∑_{p∈P} S(X(p)) and H(S(X)⊗_{P^{o}}ℤ) ≅ H(P) induced by inclusion. As in Proposition 1.3.9 the diagram S(X) is free. For this note that the existence of maxima follows from the existence of minima. Since s^{P} maps ΔX(p) to X(p), it induces a map ℤ ⊗_{P^{o}} B(P^{o}) → S(X). Regarding ℤ as a chain complex concentrated in dimension 0 this is a ZŽ–map, because X(p) is acyclic for all p. The resulting isomorphism

H_{∗}(ΔP) ≅ H(ℤ ⊗_{P^{o}} B(P^{o}) ⊗_{P^{o}} ℤ) → H(S(X) ⊗_{P^{o}} ℤ) ≅ H(P)

is easily identified with s_{∗}^{P}. □

2.3.11 Proposition. The composition

H_{∗}(ΔQ_{[k,n]}, ΔQ_{[k,n)}) ⟶^{h_{k}} H_{∗}(P V, ⋃ P A) ⟶^{s_{∗}^{A}} H_{∗}(Q_{[0,n]}, Q_{[0,n)})

is an isomorphism for k = 0 and zero for k > 0.

Proof. Consider the diagram

\xymatrix{ℂP^{k} × (ΔQ_{[k,n]}, ΔQ_{[k,n)}) \ar[rr]^{f^{k}} \ar[d]^{π} &  & (P V, ⋃ P A) \ar[d]^{s^{A}} \\ (ΔQ_{[k,n]}, ΔQ_{[k,n)}) \ar[r]^{s^{Q_{[k,n]}}} & (Q_{[k,n]}, Q_{[k,n)}) \ar[r]^{i} & (Q_{[0,n]}, Q_{[0,n)})}

where i is the inclusion map and π the projection onto the second factor. By construction of f^{k}, f^{k} [{x} × ⟨u_{0}, …, u_{q}⟩] ⊂ P u_{q}, that is s^{A}(f^{k}(x, y)) ≤ s^{Q}(y), and by Lemma 2.3.7 this implies the homotopy commutativity of the diagram.

For k = 0, π is a homeomorhism, i the identity, and h_{0} equals f_{∗}^{0} up to an isomorphism. Therefore s_{∗}^{A} ∘h_{0} is an isomorphism, because s_{∗}^{Q_{[0,n]}} is an isomorphism by Lemma 2.3.10.

For k > 0, s_{∗}^{A}(h_{k}(c)) = s_{∗}^{A}(f_{∗}^{k}([ℂP^{k}] × c)) = (i ∘ s^{Q_{[k,n]}})_{∗}(π_{∗}([ℂP^{k}] × c)) = (i ∘ s^{Q_{[k,n]}})_{∗}(0) = 0. □

Since we are concerned with the vanishing of certain intersection products, we will use the following immediate corollary.

2.3.12 Corollary. Let 0 ≤ i ≤ n, c_{i} ∈ H_{∗}(ΔQ_{[k,n]}, ΔQ_{[k,n)}), and x = ∑_{i} h_{i}(c_{i}) ∈ H_{∗} (P V, ⋃ P A). If s_{∗}^{A}(x) = 0 ∈ H_{∗}(Q_{[0,n]}, Q_{[0,n)}), then c_{0} = 0. □

Vanishing for projective arrangements

We are now ready to prove the last step in the proof of the product formula for complex projective arrangements, namely the following proposition.

2.3.13 Proposition. Let 𝕂 = ℂ and k + l < n, c ∈ H_{∗}(ΔQ_{[k,n]}, ΔQ_{[k,n)}), d ∈ H_{∗}(ΔQ_{[l,n]}, ΔQ_{[l,n)}). Then

h_{k}(c) • h_{l}(d) = 0.

We will assume 𝕂 = ℂ from now on. We will prove the proposition in three steps.

We would like to have the classes h_{k}(c) and h_{l}(d) represented by chains as much as possible in general position with respect to each other. To this end we consider an arrangement that is the union of two arrangements A and Ã with intersection posets Q and Q̃ such that Ã is in general position with respect to A (see Definition 2.3.4).

We will denote the intersection poset of the arrangement Ǎ := A ∪ Ã by Q̌ and so on. The map

\begin{matrix}σ: (Q × Q̃)_{[n,2n]} & → Q̌_{[0,n]} \\ (u, v) & ↦ u ∩ v \end{matrix}  (2.58)

is an isomorphism.

2.3.14 Proposition. In the above situation let c ∈ H_{∗}(ΔQ_{[k,n]}, ΔQ_{[k,n)}) and d ∈ H_{∗}(ΔQ̃_{[l,n]}, ΔQ̃_{[l,n)}) with k + l < n. Then

h_{k}(c) • h̃_{l}(d) = ∑_{i>0} ȟ_{i}(r_{i})

for classes r_{i} ∈ H_{∗} (ΔQ̌_{[i,n]}, ΔQ̌_{[i,n)}).

Proof. By Corollary 2.3.12 we have to show sǍ ; ∗(h_{k}(c) • h̃_{l}(d)) = 0. It will be in doing so that we employ the ideas laid out in the discussion of the real example.

We set (X, A) := ℂP^{k} × ℂP^{l} × (Δ(Q_{[k,n]} × Q̃_{[l,n]}), Δ(Q_{[k,n]} × Q̃_{[l,n]})_{[0,2n)}), D := {(x, x) ∈ P V × P V }, CD := (P V × P V) \ D and use the map g from Definition 2.3.1. We denote the diagonal map P V → P V × P V by Δ and define ḡ: (g^{−1}[D], g^{−1}[D] ∩ A) → (P V, ⋃ P A ∪ ⋃ P Ã) by Δ ∘ ḡ = g. Note that in the real example the projection of g^{−1}[D] to the order complex is the set represented by a dotted line in (2.56). We will first show h_{k}(c) • h̃_{l}(d) ∈ im ḡ_{∗} and then

sǍ ; ∗ ∘ ḡ_{∗} = 0.

There is a commutative diagram

\xymatrix{H^{∗}(P V × P V, CD) ⊗ H_{∗}(X, A) \ar[r]^{id⊗g_{∗}} \ar[d]^{g^{∗}⊗id} & {H^{∗}(P V × P V, CD)⊗ \atop H_{∗} ((P V, ⋃ P A) × (P V, ⋃ P Ã))} \ar[dd]^{⌢} \\ H^{∗}(X, X \ g^{−1}[D]) ⊗ H_{∗}(X, A) \ar[d]^{⌢} \\ H_{∗}(g^{−1}[D], g^{−1}[D] ∩ A) \ar[r]^{g_{∗}} \ar[dr]_{ḡ_{∗}} & H_{∗}(D, D ∩ (⋃ P A × P V ∪ P V × ⋃ P Ã)) \\  & H_{∗}(P V, ⋃ P A ∪ ⋃ P Ã). \ar[u]^{≅}_{Δ_{∗}}}

Regarding the existence of the cap products in this diagram and commutativity, note that we are entirely dealing with algebraic sets and polynomial maps. Now, if ϑ ∈ H^{∗}(P V × P V, CD) is the Thom class determined by ϑ ⌢ [P V × P V] = Δ_{∗}([P V]), then

h_{k}(c) • h̃_{l}(d) = Δ_{!}(h_{k}(c) × h̃_{l}(d))
= Δ_{∗}^{−1} (ϑ ⌢ (h_{k}(c) × h̃_{l}(d)))
= Δ_{∗}^{−1} (ϑ ⌢ g_{∗}([ℂP^{k}] × [ℂP^{l}] × (c × d)))
= Δ_{∗}^{−1} (g_{∗} (g^{∗}(ϑ) ⌢ ([ℂP^{k}] × [ℂP^{l}] × (c × d))))
= ḡ_{∗} (g^{∗}(ϑ) ⌢ ([ℂP^{k}] × [ℂP^{l}] × (c × d))) .

By construction of g, ḡ (x, y, ∑_{j=0}^{r} λ_{j}(u_{j}, v_{j})) ∈ u_{r} ∩ v_{r}. Firstly this implies that

g^{−1}[D] misses ℂP^{k} × ℂP^{l} × Δ(Q × Q̃)_{[0,n)}, and secondly from the reformula-

tion sǍ(ḡ(x, y, z)) ≤ σ (s^{Q×Q̃}(z)), where σ is the isomorphism from (2.58), and Lemma 2.3.7 it can be seen that the diagram

[figure: (g^{−1}[D],g^{−1}[D]∩A); π; sǍ∘ḡ; (Q̌_{[0,n]},Q̌_{[0,n)}); ((Q×Q̃)_{[n,2n]},(Q×Q̃)_{[n,2n)}); σ; (Δ(Q_{[k,n]}×Q̃_{[l,n]})\Δ(Q×Q̃)_{[0,n)},Δ(Q_{[k,n]}×Q̃_{[l,n]})_{[0,2n)}\Δ(Q×Q̃)_{[0,n)}),; s^{Q×Q̃}]

where π denotes projection onto the third factor, is homotopy commutative. The two arrows on the right hand side of the diagram should be compared to (2.57).

By Proposition 2.3.5 and because the subcomplex Δ((Q_{[k,n]}×Q̃_{[l,n]})∩(Q×Q̃)_{[0,n)}) has dimension at most n − 1 − k − l, we may assume that π[g^{−1}[D]] will not only miss this subcomplex, but any cone over it. Therefore π factorizes over the pair

(Δ(Q_{[k,n]} × Q̃_{[l,n]}) \ Δ((Q × Q̃)_{[0,n)} ∪ {(V, V)}),

Δ(Q_{[k,n]} × Q̃_{[l,n]})_{[0,2n)} \ Δ(Q × Q̃)_{[0,n)}).

This pair is homeomorphic to (Δ(Q_{[k,n]} ×Q̃_{[l,n]})_{[0,2n)} \Δ(Q×Q̃)_{[0,n)})×([0, 1), {0})

and has trivial homology. So (sǍ ∘ ḡ)_{∗} = s_{∗}^{Q×Q̃} ∘ π_{∗} = s_{∗} ∘ 0 = 0. □

2.3.15 Proposition. In the above situation let c ∈ H_{∗}(ΔQ_{[k,n]}, ΔQ_{[k,n)}) and d ∈ H_{∗}(ΔQ̃_{[l,n]}, ΔQ̃_{[l,n)}) with k + l < n. Then

h_{k}(c) • h̃_{l}(d) = 0.

Proof. We choose a hyperplane H in V in general position with respect to the arrangement Ǎ = A∪Ã and use notation as in Section 2.1. By Proposition 2.1.35 and induction on the dimension of V

i_{!}(h_{k}(c) • h̃_{l}(d)) = i_{!}(h_{k}(c)) • i_{!}(h̃_{l}(d)) = h_{k−1}^{H}(η_{∗}(c)) • h̃_{l−1}^{H}(η_{∗}(d)) = 0,

since the arrangement A^{H} is again in general position with respect to the arrangement Ã^{H} and (k − 1) + (l − 1) < n − 1. This implies h_{k}(c) • h̃_{l}(d) ∈ ker i_{!} = im ȟ_{0} by Proposition 2.1.35. But h_{k}(c) • h̃_{l}(d) ∈ ⨁_{j>0} im ȟ_{j} by Proposition 2.3.14 and hence h_{k}(c) • h̃_{l}(d) = 0. □

Proof of Proposition 2.3.13. We choose a neighbourhood U of ⋃ P A such that the inclusion (P V, ⋃ P A) → (P V, U) is a homotopy equivalence. We then choose a copy Ã of A also contained in U, in general position with respect to A and such that the diagram

\xymatrix{ℂP^{i} × (ΔQ_{[i,n]}, ΔQ_{[i,n)}) \ar[d]^{f^{i}} \ar[r]^{f̃^{i}} & (P V, ⋃ P Ã) \ar[d]^{incl.} \\ (P V, ⋃ P A) \ar[r]|{incl.} & (P V, U)}

commutes up to homotopy. Because of the commutativity of

\xymatrix{H_{∗}(P V, ⋃ P A) ⊗ H_{∗}(P V, ⋃ P A) \ar[r]^{•} & H_{∗}(P V, ⋃ P A) \ar[d]_{_{incl_{∗}}}^{≅} \\ H_{∗}(ΔQ_{[k,n]}, ΔQ_{[k,n)}) ⊗ H_{∗}(ΔQ_{[l,n]}, ΔQ_{[l,n)}) \ar[u]_{h_{k}⊗h_{l}} \ar[d]^{h_{k}⊗h̃_{l}} & H_{∗}(P V, U) \\ H_{∗}(P V, ⋃ P A) ⊗ H_{∗}(P V, ⋃ P Ã) \ar[r]^{•} & H_{∗}(P V, ⋃ P A ∪ ⋃ P Ã) \ar[u]^{_{incl_{∗}}}}

the result follows from Proposition 2.3.15 □

Proof of Theorem 2.2.1. The theorem follows directly from Proposition 2.3.13 and Proposition 2.2.10. □

2.4 Projective c-arrangements

Descriptions of the cohomology ring of the complement of a linear arrangement in terms of generators and relations go back to Arnol’d [Arn69] who gave such a description for the classifying space of the coloured braid group, which is the complement of a complex hyperplane arrangement. He also conjectured a similar formula for general complex hyperplane arrangements, later to be proved by Orlik and Solomon [OS80] (see Remark 2.4.4). Since then several such results on other classes of linear arrangements have been obtained. The approach most useful to us is that of Yuzvinsky, who derived from the complex case of Theorem 2.2.3 (with rational coefficients) a description in terms of generators and relations of the cohomology ring of the complement of a complex linear arrangement with geometric intersection lattice [Yuz99]. Generalizations of his results to real ≥ 2arrangements and integral coefficients have been stated in [dLS01].

A presentation of the cohomology ring

We will now use a route similar to Yuzvinsky’s to obtain from Theorem 2.2.1 a simple description of the cohomology of the complement of a complex projective c-arrangement. These probably form the simplest class of arrangements that still yield a proper generalization of the classical result on complex hyperplane arrangements in this way. The result presented in this section complements results of Feichtner and Ziegler in [FZ00].

2.4.1 Definition. For a positive integer c, we call A a c-arrangement, if every A ∈ A is a subspace of codimension c and d(q) is an integral multiple of c for every q ∈ Q.

2.4.2 Definition. We call a subset M of A independent, if n − d (⋂ M) = ∑_{A∈M}(n − d(A)), dependent, if it is not independent, and minimally dependent, if it is dependent but all of its proper subsets are independent.

2.4. Projective c-arrangements 69

We will assume 𝕂 = ℂ in this section. Our goal is the following.

2.4.3 Theorem. Let A be a complex c-arrangement, |A| − 1 =: t ≥ 0, A = {A_{0}, …, A_{t}}. Let R be the free graded commutative (in the graded sense) ring over the set of generators {x} ∪ {y_{i}: 1 ≤ i ≤ t} with |x| = 2, |y_{i}| = 2c − 1. Let I be the ideal generated by

{∑_{j=0}^{r}(−1)^{j}y_{i_{0}} ⋯ ŷ_{i_{j}} ⋯ y_{i_{r}}: i_{0} < ⋯ < i_{r}, {A_{i_{j}}} is minimally dependent.}

∪ {y_{i_{1}} ⋯ y_{i_{r}}: i_{1} < ⋯ < i_{r}, {A_{0}} ∪ {A_{i_{j}}} is minimally dependent. } ∪ {x^{c}} .

The map

π: R → H^{∗} (P V \ ⋃ P A) ,
x ↦ P (h_{n−1}([⟨V ⟩])),
y_{i} ↦ P (h_{n−c}([⟨A_{i}, V ⟩ − ⟨A_{0}, V ⟩])),

where P: H_{∗}(P V, ⋃ P A) ⟶^{≅} H^{∗} (P V \ ⋃ P A) denotes Poincaré duality, is an epimorphism and ker π = I.

We now fix the arrangement A = {A_{0}, …, A_{t}}.

2.4.4 Remark. For c = 1 the complement P V \ ⋃ P A can be regarded as the complement in the affine space P V \ P A_{0} of the linear hyperplane arrangement A′ := {P A_{i} \ P A_{0} : 1 ≤ i ≤ t}. In this case, the generator x and the corresponding relation can be omitted.

If A_{0} is in general position with respect to A\{A_{0}}, the second kind of generators does not occur. This is in particular the case if the arrangement A′ is central, i.e. if ⋂ A′ ≠ Ø. In this case the theorem reduces to the description of the cohomology ring of the complement of A′ given by Orlik and Solomon.

The atomic complex

We now turn to the proof of the theorem. When using simplicial chain complexes, we will always use the complex of non-degenerate simplices and view it as the complex of all simplices modulo degenerate simplices if necessary.

2.4.5 Definition. For an integer k with 0 ≤ k ≤ n, we define S_{k} to be the simplicial complex which has the vertex set {0, …, t} and as simplices the sets I ⊂ {0, …, t} with d (⋂_{i∈I} A_{i}) ≥ k. This is the atomic complex of Q_{[k,n]}. We also define D^{k} to be the reduced ordered (using the natural order of {0, …, t}) simplicial chain complex of S_{k} shifted by one, i.e. D_{r}^{k} = C̃_{r−1}(S_{k}) and in particular D_{0}^{k} ≅ ℤ generated by the empty simplex.

As is well known, the atomic complex and the order complex, of Q_{[k,n)} in this case, are homotopy equivalent. We describe a homotopy equivalence to fix a concrete isomorphism between their homology groups. Before doing this, we state a useful lemma.

2.4.6 Lemma. Let P_{0}, P_{1} be posets, P′_{i} ⊂ P_{i}. If f, g: (P_{0}, P′_{0}) → (P_{1}, P′_{1}) are order preserving functions such that f(p) ≤ g(p) for all p ∈ P_{0}, then the maps f, g: (ΔP_{0}, ΔP′_{0}) → (ΔP_{1}, ΔP′_{1}) are homotopic.

Proof. The map H: {0, 1}×P_{0} → P_{1} defined by H(0, x) := f(x), H(1, x) := g(x) is order preserving and hence yields the desired homotopy

I × (ΔP_{0}, ΔP′_{0}) ≈ (Δ({0, 1} × P_{0}), Δ({0, 1} × P′_{0})) ⟶^{H} (ΔP_{1}, ΔP′_{1}),

where we view {0, 1} as a poset. □

2.4.7 Remark. This lemma is a special case of [Seg68, Prop. 2.1] which is proved in the same way.

2.4.8 Definition and Proposition. We denote the face poset of S_{k} by F S_{k}, but order it by M ≤ M′ if M′ is a face of M, that is if M′ ⊂ M. We also set F̃S_{k} := F S_{k} ∪ {Ø}. The map

s: (F̃S_{k}, F S_{k}) → (Q_{[k,n]}, Q_{[k,n)})
M ↦ ⋂ {A_{i}: i ∈ M}

is then order preserving and moreover satisfies s(M ∧ M′) = s(M ∪ M′) = s(M) ∩ s(M′) = s(M) ∧ s(M′), if one side, and therefore the other, exists.

With these definitions, the map s: (ΔF̃S_{k}, ΔF S_{k}) → (ΔQ_{[k,n]}, ΔQ_{[k,n)}) is a homotopy equivalence.

2.4.9 Remark. ΔF S_{k} is the barycentric subdivision of S_{k}, and ΔF̃S_{k} is a cone over ΔF S_{k}.

Proof. We define an order preserving map

r: (Q_{[k,n]}, Q_{[k,n)}) → (F̃S_{k}, F S_{k}),
q ↦ {i: A_{i} ⊃ q} .

We have s(r(q)) ≥ q for q ∈ Q_{[k,n]} and r(s(i)) ≤ i for i ∈ F̃S_{k}. Hence, by the preceding lemma r is a homotopy inverse to s, when both maps are regarded as simplicial maps between order complexes. □

2.4.10 Definition and Proposition. We define chain maps

f^{k}: D_{r}^{k} → C_{r}(ΔQ_{[k,n]}, ΔQ_{[k,n)})
⟨i_{1}, …, i_{r}⟩ ↦ ⟨A_{i_{1}}, V ⟩ ˆ× ⋯ ˆ×⟨A_{i_{r}}, V ⟩.

For r = 0 this is to be understood as f^{k}(⟨⟩) = ⟨V ⟩.

2.4.11 Notation. To simplify the following calculations, we set set

α_{i} := ⟨A_{i}, V ⟩ ∈ C_{1}(ΔQ_{[n−c,n]})

and sometimes write the multiplication ˆ× as juxtaposition.

Proof. To see that f^{k} is well-defined, we have to check that the right hand side is in C_{∗}(ΔQ_{[k,n]}). But d (A_{i_{1}} ∧ ⋯ ∧ A_{i_{r}}) ≥ k by definition of S_{k} and hence D_{r}^{k}.

To see that f^{k} is a chain map, we calculate

𝔡(f^{k}(⟨i_{1}, …, i_{r}⟩)) = ∑_{j=1}^{r}(−1)^{j+1}⟨A_{i_{1}}, V ⟩ ˆ× ⋯ ˆ× 𝔡⟨A_{i_{j}}, V ⟩ ˆ× ⋯ ˆ×⟨A_{i_{r}}, V ⟩

= ∑_{j=1}^{r}(−1)^{j+1}α_{i_{1}} ⋯ α_{i_{j−1}}⟨V ⟩α_{i_{j+1}} ⋯ α_{i_{r}}

− ∑_{j=1}^{r}(−1)^{j+1}α_{i_{1}} ⋯ α_{i_{j−1}}⟨A_{i_{j}}⟩α_{i_{j+1}} ⋯ α_{i_{r}}

= ∑_{j=1}^{r}(−1)^{j+1}α_{i_{1}} ⋯ α_{i_{j−1}} α̂_{i_{j}}α_{i_{j+1}} ⋯ α_{i_{r}}

− ∑_{j=1}^{r}(−1)^{j+1}α_{i_{1}} ⋯ α_{i_{j−1}}⟨A_{i_{j}}⟩α_{i_{j+1}} ⋯ α_{i_{r}}.

The first summand equals f^{k}(𝔡⟨i_{1}, …, i_{r}⟩). Since ⟨A_{i}⟩ ∈ ℂ_{0}(Q_{[n−c,n)}), the second summand is in C_{∗}(ΔQ_{[k,n−c]}) ⊂ C_{∗}(ΔQ_{[k,n)}). □

2.4.12 Proposition. The induced maps f_{∗}^{k}: H(D^{k}) → H(ΔQ_{[k,n]}, ΔQ_{[k,n)}) are isomorphisms.

Proof. Defining

f̄^{k}: D_{r}^{k} → C_{r}(ΔF̃S_{k}, ΔF S_{k})
⟨i_{1}, …, i_{r}⟩ ↦ ⟨{i_{1}} , Ø⟩ ˆ× ⋯ ˆ×⟨{i_{r}} , Ø⟩

the diagram

\xymatrix{ & H_{r}(D^{k}) \ar[d]_{f̄_{∗}^{k}} \ar[dl]_{f_{∗}^{k}} \ar@{-->}[dr]^{_{sd_{∗}}}_{≅} \\ H_{r}(ΔQ_{[k,n]}, ΔQ_{[k,n)}) & H_{r}(ΔF̃S_{k}, ΔF S_{k}) \ar[l]_{s_{∗}}^{≅} \ar[r]^{𝔡}_{≅} & H̃_{r−1}(ΔF S_{k})}

commutes, where sd is the barycentric subdivision map C̃_{∗}(S_{k}) → C̃_{∗}(ΔF S_{k}). The connecting homomorphism is an isomorphism, because F̃S_{k} has the maximum Ø. The map s_{∗} is an isomorphism because of Proposition 2.4.8. It follows that f_{∗}^{k} is an isomorphism. □

Proof of the presentation

The chain maps f^{k} would be more useful in a situation in which the chains ⟨A_{i}, V ⟩ are cycles. For example, think of affine arrangements, where (ΔQ_{[k,n]}, ΔQ_{[k,n)} ∪ ΔQ_{(k,n]}) takes the place of (ΔQ_{[k,n]}, ΔQ_{[k,n)}). In our situation they are not. The chains ⟨A_{i}, V ⟩ − ⟨A_{j}, V ⟩ however are cycles, we will therefore replace the maps f^{k} by the following maps.

2.4.13 Definition and Proposition. For a c-arrangement A, we define chain maps

g^{k}: D_{r}^{k} → C_{r}(ΔQ_{[k,n]}, ΔQ_{[k,n)})
⟨i_{1}, …, i_{r}⟩ ↦ \begin{cases}(⟨A_{i_{1}}, V ⟩ − ⟨A_{0}, V ⟩) ˆ× ⋯ ˆ×(⟨A_{i_{r}}, V ⟩ − ⟨A_{0}, V ⟩), & r = a, \\ 0, & r ≠ a, \end{cases}

where a is defined by n − (a + 1)c < k ≤ n − ac.

Proof. We check that g^{k} is a well-defined chain map. For r > a we have C_{r}(ΔQ_{[k,n]}, ΔQ_{[k,n)}) ≅ 0, since n − k < (a + 1)c ≤ rc. So we just have to show that g^{k}(⟨i_{1}, …, i_{a}⟩) is a cycle in C_{a}(ΔQ_{[k,n]}, ΔQ_{[k,n)}). This is true, because each ⟨A_{i}, V ⟩ − ⟨A_{0}, V ⟩ is a cycle in C_{1}(ΔQ_{[n−c,n]}, ΔQ_{[n−c,n)}) and n − ac ≥ k. □

2.4.14 Proposition. The maps f^{k} and g^{k} are chain homotopic.

Proof. We define

K: D_{r}^{k} → C_{r+1}(ΔQ_{[k,n]}, ΔQ_{[k,n)})
⟨i_{1}, …, i_{r}⟩ ↦ \begin{cases}f^{k} (⟨0, i_{1}, …, i_{r}⟩) , & r < a, \\ 0, & r ≥ a. \end{cases}

The right hand side is well defined, because for r < a we have

d(A_{0} ∩ A_{i_{1}} ∩ ⋯ ∩ A_{i_{r}}) ≥ n − (r + 1)c ≥ n − ac ≥ k.

We calculate K𝔡 + 𝔡K.

For r < a:

(K𝔡 + 𝔡K)⟨i_{1}, …, i_{r}⟩

= f^{k} (∑_{j=1}^{r}(−1)^{j+1}⟨0, i_{1}, …, î_{j}, …, i_{r}⟩) + 𝔡f^{k} (⟨0, i_{1}, …, i_{r}⟩)

= f^{k} (∑_{j=1}^{r}(−1)^{j+1}⟨0, i_{1}, …, î_{j}, …, i_{r}⟩ + 𝔡⟨0, i_{1}, …, i_{r}⟩)

= f^{k}(⟨i_{1}, …, i_{r}⟩) = (f^{k} − g^{k})⟨i_{1}, …, i_{r}⟩.

For r = a: We first calculate

g^{k}(⟨i_{1}, …, i_{a}⟩) = (α_{i_{i}} − α_{0}) ⋯ (α_{i_{a}} − α_{0})

= α_{i_{1}} ⋯ α_{i_{a}} − ∑_{j=1}^{a} α_{i_{1}} ⋯ α_{i_{j−1}}α_{0}α_{i_{j+1}} ⋯ α_{i_{a}}

= α_{i_{1}} ⋯ α_{i_{a}} + ∑_{j=1}^{a}(−1)^{j}α_{0}α_{i_{1}} ⋯ α̂_{i_{j}} ⋯ α_{i_{a}}

and with this

(K𝔡 + 𝔡K)⟨i_{1}, …, i_{a}⟩ = f^{k} (∑_{j=1}^{a}(−1)^{j+1}⟨0, i_{1}, …, î_{j}, …, i_{a}⟩)

= ∑_{j=1}^{a}(−1)^{j+1}α_{0}α_{i_{1}} ⋯ α̂_{i_{j}} ⋯ α_{i_{a}}

= (f^{k} − g^{k})⟨i_{1}, …, i_{a}⟩.

For r > a we have (K𝔡 + 𝔡K)⟨i_{1}, …, i_{r}⟩ = 0 = (f^{k} − g^{k})⟨i_{1}, …, i_{r}⟩, since C_{r}(ΔQ_{[k,n]}, ΔQ_{[k,n)}) ≅ 0 as noted before. □

2.4.15 Proposition. The map π is surjective.

Proof. By Proposition 2.4.12, Proposition 2.4.14, and of course Proposition 2.1.25, H^{∗}(P V \ ⋃ P A) is additively generated by the elements P (h_{k} ([g^{k}⟨i_{i}, …, i_{r}⟩])) with k ≤ n − rc. By Theorem 2.2.1

P (h_{k} ([g^{k}⟨i_{i}, …, i_{r}⟩]))
= P (h_{k}([(α_{i_{1}} − α_{0}) ˆ× ⋯ ˆ×(α_{i_{r}} − α_{0})]))
= P (h_{n−1}([⟨V ⟩]))^{n−k−rc}P (h_{n−c}([α_{i_{1}} − α_{0}])) ⋯ P (h_{n−c}([α_{i_{r}} − α_{0}]))
= π(x)^{n−k−rc}π(y_{i_{1}}) ⋯ π(y_{i_{r}}).

This shows that π is surjective. □

2.4.16 Proposition. I ⊂ ker π.

Proof. First of all

π(x^{c}) = P (h_{n−1}([⟨V ⟩]))^{c}
= P (h_{n−c}([⟨V ⟩]))
= P (h_{n−c}([𝔡⟨A_{0}, V ⟩])) = P (h_{n−c}(0)) = 0.

If {A_{i_{0}}, …, A_{i_{r}}} is minimally dependent, then d (⋂_{j} A_{i_{j}}) = n − rc and

0 = P (h_{n−rc}(g_{∗}^{n−rc}([𝔡⟨i_{0}, …, i_{r}⟩])))

= (P ∘ h_{n−rc} ∘ g_{∗}^{n−rc}) ([∑_{j=0}^{r}(−1)^{j}⟨i_{0}, …, î_{j}, …, i_{r}⟩])

= π (∑_{j=0}^{r}(−1)^{j}y_{i_{0}} ⋯ ŷ_{i_{j}} ⋯ y_{i_{r}} )

and similarly if {A_{0}, A_{i_{1}}, …, A_{i_{r}}} is minimally dependent, then

0 = P (h_{n−rc}(g_{∗}^{n−rc}([𝔡⟨0, i_{1}, …, i_{r}⟩])))

= (P ∘ h_{n−rc} ∘ g_{∗}^{n−rc}) ([⟨i_{1}, …, i_{r}⟩ + ∑_{j=1}^{r}(−1)^{j}⟨0, i_{1}, …, î_{j}, …, i_{r}⟩])

= π (y_{i_{1}} ⋯ y_{i_{r}})

as claimed. □

2.4.17 Lemma. If i_{0} < ⋯ < i_{r} and {A_{i_{j}}} is dependent, then y_{i_{0}} ⋯ y_{i_{r}} ∈ I and ∑_{j}(−1)^{j}y_{i_{0}} ⋯ ŷ_{i_{j}} ⋯ y_{i_{r}} ∈ I.

Proof. Let {A_{i_{0}}, …, A_{i_{r}}} be dependent. To show y_{i_{0}} ⋯ y_{i_{r}} ∈ I we may assume that the set is minimally dependent. Then ∑_{j}(−1)^{j}y_{i_{0}} ⋯ ŷ_{i_{j}} ⋯ y_{i_{r}} ∈ I and

y_{i_{0}} (∑_{j}(−1)^{j}y_{i_{0}} ⋯ ŷ_{i_{j}} ⋯ y_{i_{r}}) = y_{i_{0}} ⋯ y_{i_{r}} since y_{0}^{2} = 0.

For the second part of the lemma we may assume that {A_{i_{j}}: j ≤ s} is minimally dependent. Then

∑_{j}(−1)^{j}y_{i_{0}} ⋯ ŷ_{i_{j}} ⋯ y_{i_{r}} =

\underbrace{(∑_{j=0}^{s}(−1)^{j}y_{i_{0}} ⋯ ŷ_{i_{j}} ⋯ y_{i_{s}} )}_{∈I} y_{i_{s+1}} ⋯ y_{i_{r}}

+ \underbrace{y_{i_{0}} ⋯ y_{i_{s}}}_{∈I} ∑_{j=s+1}^{r}(−1)^{j}y_{i_{s+1}} ⋯ ŷ_{i_{j}} ⋯ y_{i_{r}} ∈ I

as claimed. □

2.4.18 Proposition. ker π ⊂ I.

Proof. Let z ∈ ker π. We want to show z ∈ I. We may assume that z is a linear combination of elements x^{s}y_{i_{1}} ⋯ y_{i_{r}} with 0 ≤ s < c, i_{1} < ⋯ < i_{r} and {A_{i_{1}}, …, A_{i_{r}}} independent. Since π(x^{s}y_{i_{1}} ⋯ y_{i_{r}}) ∈ im(P ∘ h_{n−cr−s}) and r and s are determined by cr + s, we may assume z to be homogenous in r and s, i.e. z = x^{s} ∑_{i_{1}<···<i_{r}} λ_{i}y_{i_{1}} ⋯ y_{i_{r}}. We set k := n − cr − s. The chain

z′ := ∑_{i_{1}<···<i_{r}} λ_{i}⟨i_{1}, …, i_{r}⟩ + ∑_{i_{1}<···<i_{r}} λ_{i} ∑_{j=1}^{r}(−1)^{j}⟨0, i_{1}, …, î_{j}, …, i_{r}⟩

is a cycle in D_{r}^{k} (the second summand is a cone over the boundary of the first summand), 0 = π(z) = (P ∘ h_{k}) (g_{∗}^{k} ([z′])), and therefore [z′] = 0 by Proposition 2.4.12 and Proposition 2.4.14, i.e. z′ is a boundary in D^{k}, which means that there exist μ_{i}, ν_{i} such that

z′ = 𝔡 ( ∑_{i_{1}<···<i_{r}} μ_{i}⟨0, i_{1}, …, i_{r}⟩ + ∑_{i_{0}<···<i_{r}} ν_{i}⟨i_{0}, …, i_{r}⟩)

and with d (⋂_{j} A_{i_{j}} ∩ A_{0}) ≥ k and therefore {A_{0}} ∪ {A_{i_{j}}} dependent for μ_{i} ≠ 0 and {A_{i_{j}}} dependent for ν_{i} ≠ 0. Comparing coefficients and sorting the simplices by whether the first vertex is 0 yields

z = ∑_{i} μ_{i}y_{i_{1}} ⋯ y_{i_{r}} + ∑_{i} ν_{i} ∑_{j}(−1)^{j}y_{i_{0}} ⋯ ŷ_{i_{j}} ⋯ y_{i_{r}} ∈ I

as claimed. □

This completes the proof of Theorem 2.4.3 □ 76 2. Linear and related arrangements

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Zusammenfassung

Diese Dissertationsschrift befasst sich mit Homotopie- und Homologieeigenschaften von Arrangements. Ein Arrangement in einem topologischen Raum X ist eine endliche Menge A von Unterräumen von X. Ein Ziel beim Studium von Arrangements ist die Beschreibung der Vereinigung ⋃ A und des Komplements X \ ⋃ A.

Das Hauptergebnis dieser Arbeit ist die Beschreibung des Kohomologierings des Komplements eines Arrangements von linearen Unterräumen eines komplexen projektiven Raumes. Da die additive Struktur des Rings bereits durch Goresky und MacPherson bestimmt wurde, kommt es hier auf die Beschreibung der Produkte an. Dies geschieht durch eine Formel wie sie von Yuzvinsky für komplexe lineare Arrangements angegeben und für rationale Koeffizienten bewiesen wurde. Die Darstellung der Formel ist wie in einer gemeinsamen Arbeit von de Longueville und dem Autor, in der die Formel für den linearen Fall für ganzzahlige Koeffizienten bewiesen und auf gewisse reelle Arrangements verallgemeinert wurde.

Das erste Kapitel behandelt Arrangements in allgemeinen topologischen Räumen. Es werden zuerst kurz Ergebnisse über Diagramme von Räumen dargestellt, die von Ziegler und Živaljević als für das Studium von Homotopieeigenschaften von Arrangements als nützlich erkannt wurden. Wir entwickeln dann eine analoge Theorie von Diagrammen von Kettenkomplexen, um beim Studium von Homologieeigenschaften mehr Freiräume zu haben. Dabei tritt zentral eine Spektralsequenz in Erscheinung, die wir im letzten Abschnitt des ersten Kapitels nutzen, um für Arrangements in Mannigfaltigkeiten eine Produktformel für den Kohomologierings des Komplements herzuleiten. Dadurch, dass sie so allgemein gilt, kann sie aber den Ring nicht vollständig beschreiben. Sie ist gradiert in dem Sinne, dass sie Produkte nur bis auf Terme niedrigerer Filtrierungsstufen im Sinne der Spektralsequenz bestimmt.

Das zweite Kapitel widmet sich linearen Arrangements, affinen wie projektiven. Zunächst werden die Methoden des ersten Kapitels angewandt, um diverse Homologie- und einige Homotopieformeln auf einheitliche Weise zu gewinnen. Dann werden gradierte Produktformeln für affine und projektive Arrangements bewiesen. Es wird gezeigt, dass sich aus ihnen induktiv exakte Formeln gewinnen lassen, wenn sich das Verschwinden von Produkten in gewissen Fällen sicherstellen lässt. Dies ist für affine Arrangements durch ein einfaches geometrisches Argument zu bewerkstelligen. Für projektive Arrangements ist es deutlich schwieriger, dies ist das technische Herz dieser Arbeit. Schließlich wird die Produktformel für projektive Arrangements im Geiste Yuzvinskys genutzt, um aus ihr eine Präsentierung des Kohomologierings nach Orlik-Solomon-Art für eine spezielle Klasse komplexer projektiver Arrangements abzuleiten.