## MAS435 Algebraic Topology

Note: Information for future academic years is provisional. Timetable information and teaching staff are especially likely to change, but other details may also be altered, some courses may not run at all, and other courses may be added.

 Both semesters, 2022/23 20 Credits Lecturer: Prof Neil Strickland Home page Timetable Reading List Aims Outcomes Assessment

In this course, we will study geometric objects from a topological point of view. This means we will be not be interested in the exact shape of a space or the distances between points, but rather the properties that are preserved under stretching. We will show how to capture these properties purely in terms of open sets, giving the notion of topological space. Our examples will include balls, spheres, surfaces, the n-holed torus, the Möbius strip, the Klein bottle, other surfaces, knots, projective spaces.... We will define what it means for two spaces to be homeomorphic, but the focus will be on properties preserved by much more radical sorts of change (i.e., homotopy equivalence).

We will study a number of algebraic invariants (i.e. ways to associate algebraic objects to topological ones in which homotopy equivalent spaces give isomorphic algebraic objects). For each one, we need to show how to construct the invariant and how to calculate it. It then follows that if the algebraic objects associated to two spaces are not isomorphic, then the spaces are not homotopy equivalent.
The first invariant we consider is the fundamental group: this is a group constructed from a space by looking at loops in the space. It captures geometry in a rather accessible way, but is generally hard to calculate. On the other hand, the homology groups capture higher dimensional information, and are easier to calculate but harder to define. Once these invariants are defined we will calculate them for a range of spaces and give a variety of applications: Brouwer and Lefschetz fixed point theorems, hairy ball theorem, Ham Sandwich Theorem, Football Squashing Theorem, proof of the Fundamental Theorem of Algebra,

Note that one of the suggested books is available online free from http://pi.math.cornell.edu/~hatcher/AT/ATpage.html

Prerequisites: MAS220 (Algebra); MAS331 (Metric Spaces)
No other modules have this module as a prerequisite.

## Outline syllabus

Part A

1. Introduction and motivation
2. Simplicial complexes
3. Topological spaces
4. Homeomorphism
5. Paths
6. Categories and functors
7. Cutting invariants
8. Constructing new spaces
9. The Hausdorff property, and compactness
10. Homotopy
11. The fundamental group

Part B

12. Covering maps
13. Applications of the fundamental group
14. Neighbourhoods of subcomplexes
15. The van Kampen Theorem
16. Abelian groups
17. Chain complexes and homology
18. The chain complex of a simplicial complex
19. Chain homotopy
20. The Snake Lemma
21. The Mayer-Vietoris Theorem
22. Subdivision
23. Simplicial approximation

## Aims

• To teach the basic ideas of topological spaces, the fundamental group and homology.
• To illustrate these ideas by reference to a range of examples, including surfaces.
• To show how to calculate fundamental groups, chain complexes and homology of various topological spaces.

## Learning outcomes

• Understand the idea of a topological space.
• Understand the idea of a continuous map and a homeomorphism between topological spaces.
• Understand the idea of a homotopy between two maps, and a homotopy equivalence between two spaces.
• Understand the definition of the fundamental group and prove its basic properties.
• Calculate the fundamental group of some simple spaces such as Rn, balls, spheres, projective space, n-holed torus, Klein bottle and other surfaces.
• Understand the proofs of the Fundamental Theorem of Algebra and the Brouwer Fixed Point Theorem using the fundamental group.
• Understand the definition of simplicial homology.
• Calculate the homology of some simple spaces such as spheres, products of spheres and surfaces.
• Understand and apply the Lefschetz Fixed Point Theorem.

40 lectures, no tutorials

## Assessment

There will be eight homework assignments in each semester. The best five in each semester will each count for 2% of the overall course grade, making 20% in total. The remaining 80% will be based on a final exam. This will be an traditional closed book exam with a length of 2.5 hours. There will be five questions, and you should attempt four of them. If you attempt all five, then your best four will be counted.