Cohomology rings of arrangements

You can obtain my thesis


(Please also consult the introduction from the thesis itself.)

This thesis is concerned with homotopy and homology properties of subspace arrangements. An arrangement A in a topological space X is a finite set of subspaces of X. A goal in the study of arrangements is the description of the union and the complement of A.

The main result of this work is the description of the cohomology ring of the complement of an arrangement of linear subspaces of a complex projective space. Since the additive structure of the ring has been determined by Goresky and MacPherson, this amounts to determining cup products. This is done by a formula of the kind which for affine arrangements has been given and proved for rational coefficients by Yuzvinsky. It is presented in a form in which it has in the affine case been proved for integral coefficients and generalized to certain real arrangements by de Longueville and the author.

The first chapter is concerned with arrangements in topological spaces in general. It starts with a brief presentation of results on diagrams of spaces that have been seen to be useful in the study of homotopy properties of arrangements by Ziegler and Živaljević. We then develop an analogous theory of diagrams of chain complexes which gives some additional flexibility in the study of homology properties of arrangements. This introduces the spectral sequence that we use in the last section of the first chapter to derive a product formula for the cohomology ring of an arrangement in a manifold. Due to its generality this formula cannot describe the ring completely. It is graded in the sense that it determines products only up to terms of lower degree in the defining filtration of the spectral sequence.

The second chapter deals with linear arrangements, affine as well as projective. The methods of the first chapter are applied and yield a variety of homology formulas and some homotopy formulas. Graded product formulas for affine and projective arrangements are proved. It is shown that they allow inductive proofs of exact formulas if products vanish in certain cases. The necessary vanishing result for affine arrangements is obtained by a simple geometric argument. The corresponding result for projective arrangements is considerably harder. Its proof is the technical heart of this work. Finally we use, in the spirit of Yuzvinsky, the product formula for projective arrangements to derive from it presentations of the cohomology rings of a certain special class of complex projective arrangements, which generalize the description of the cohomology rings of hyperplane arrangements by Orlik-Solomon rings.

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