School of Mathematics and Statistics (SoMaS)

MAS6370 Algebraic Topology I

In this course, we will study spaces from a topological point of view. This means we will be interested in some notion of the "shape" of a space rather than distances between points, so the emphasis will no longer be on metrics. We will show how to generalise the notion of metric space to achieve this, giving the notion of topological space. Our examples will include balls, spheres, 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, and introduce the more subtle and expressive notion of homotopy equivalence, with some interesting examples. We will study two methods for using algebra to analyse the properties of a space: the fundamental group, and homology. For any space we will define a group, called the fundamental group, which is a beautiful and powerful way of using algebra to detect topological features of spaces; for example we can sometimes use the fundamental group to check whether two spaces are homotopy equivalent. We will calculate the fundamental groups of a number of spaces and give some applications, including a proof of the Fundamental Theorem of Algebra, and the classification of surfaces. In the second part of the course we will study homology groups, which give a more tractable method than homotopy groups for studying higher-dimensional properties of spaces.

There are no prerequisites for this module.
No other modules have this module as a prerequisite.

Outline syllabus

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

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

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.

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.