MAS435 Algebraic Topology

Both semesters, 2021/22 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).

To each space X we will associate a sequence of abelian groups Hn(X), called homology groups. Early in the course we will give an informal introduction which will be enough to understand H0(X) and H1(X) for simple spaces X that can be drawn in the plane. Over the rest of the course we will carry out the substantial technical work needed to define Hn(X) in general and prove its properties. In particular, if X and Y are homotopy equivalent then Hn(X) and Hn(Y) are isomorphic. It follows that if X and Y do not have isomorphic homology, then they are not homotopy equivalent (and so are certainly not homeomorphic). We will develop methods to calculate Hn(X) for a range of interesting spaces X, and thereby prove various non-homeomorphism results.

We will also prove several general theorems that depend on homology groups: the Brouwer Fixed Point theorem, the Fundamental Theorem of Algebra, the Borsuk-Ulam Theorem, the Jordan Curve Theorem and Invariance of Domain.


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

Office hours

Office hours will be 12-1 on Fridays. I will expect people to come to my office (Hicks J26) by default, but you can email me to arrange an online meeting if you prefer.


Aims

  • To teach the basic ideas of topological spaces, including compactness, the Hausdorff property, subspaces, products, coproducts and quotients.
  • To define singular homology groups and prove their main properties
  • To calculate homology groups of a range of interesting spaces.
  • To prove some general theorems in topology that rely on homology groups.

Learning outcomes

  • Understand the idea of a topological space.
  • Understand the idea of a continuous map and a homeomorphism between topological spaces.
  • Understand compactness and the Hausdorff property.
  • 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.

Reading list

Type Author(s) Title Library Blackwells Amazon
B Armstrong Basic Topology 513.83 (A) Blackwells Amazon
B Fulton Algebraic Topology: A First Course 514.2 (F) Blackwells Amazon
B Hatcher Algebraic Topology [free download from author's webpage] 513.83 (H) Blackwells Amazon

(A = essential, B = recommended, C = background.)

Most books on reading lists should also be available from the Blackwells shop at Jessop West.