Discrete Geometry Group

Topology II
Summer Term 2008

Times and places

Carsten Schultz: Wed 12.30-14, MA 851; Fr 10.00-11.30, MA 648.

Tutorials by Anton Dochtermann: Fr 12.15-13.45, MA 642.

Overview

We cover some topics in algebraic topology: We start with the theory of covering spaces, which is closely related to the fundamental group of a space. We then develop singular homology and cohomology theory, including products and duality in manifolds. Finally we take a look at cohomological properties of fibre bundles, in particular vector bundles. We will build on the overview given in the topology class in the winter term, but also try to keep the dependence on it small.

Problem sets

• Problem Set 1
• Problem Set 2
• Problem Set 3
• Problem Set 4
• Problem Set 5
• Problem Set 6
• Problem Set 7
• Problem Set 8
• Problem Set 9
• Problem Set 10
• Problem Set 11

There are no classes in the week starting May 26. If you want to use this time to review what has happened over the last weeks, these sugestions might be helpful.

More detailed contents

Given an infinite amount of time, we would cover chapters III to VI of [Bre93], embellishing chapter VI with material from [MS74]. Since we are given one semester, we will shorten this appropriately.

Covering Spaces

 

• We classify covering spaces of a given space and describe their deck transformation groups. We distinguish regular covering spaces.
• A covering spaces is a principal G-bundles for a discrete group G. We derive a classification for those and mention their classifying spaces.

Homology

 

• We present the Eilenberg–Steenrod axioms for homology and prove that all homology theories satisfying the axioms agree on CW-spaces.
• We define singular homology theory and prove that it satisfies the axioms. We also define the homology cross product.

Cohomology

 

• We define (singular) cohomology and products: The cohomology cross product, the cup product, and the cap product.
• We also define (although probably later in the course) the Steenrod cohomology operations.

Manifolds

 

• Poincaré–Alexander–Lefschetz duality.

Vector Bundles

 

• Vector bundles.
• The Thom isomorphism.
• Stiefel–Whitney classes.
• The Euler class.
• Classifying spaces for vector bundles.

References