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MA3H6 Algebraic Topology

Lecturer: David Mond

Term(s): Term 2

Status for Mathematics students: List A

Commitment: 30 hours

Assessment: 3 hour examination (85%), assessed work (15%)

Prerequisites: MA3F1 Introduction to Topology

Prerequisite for: MA4J7 Cohomology and Poincaré Duality

Leads To: MA4A5 Algebraic Geometry, MA5Q6 Graduate Algebra

Content: Algebraic topology is concerned with the construction of algebraic invariants (usually groups) associated to topological spaces which serve to distinguish between them. Most of these invariants are ``homotopy'' invariants. In essence, this means that they do not change under continuous deformation of the space and homotopy is a precise way of formulating the idea of continuous deformation. This module will concentrate on constructing the most basic family of such invariants, homology groups, and the applications of these homology groups.

The starting point will be simplicial complexes and simplicial homology. An n-simplex is the n-dimensional generalisation of a triangle in the plane. A simplicial complex is a topological space which can be decomposed as a union of simplices. The simplicial homology depends on the way these simplices fit together to form the given space. Roughly speaking, it measures the number of p-dimensional "holes'' in the simplicial complex. For example, a hollow 2-sphere has one 2-dimensional hole, and no 1-dimensional holes. A hollow torus has one 2-dimensional hole and two 1-dimensional holes. Singular homology is the generalisation of simplicial homology to arbitrary topological spaces. The key idea is to replace a simplex in a simplicial complex by a continuous map from a standard simplex into the topological space. It is not that hard to prove that singular homology is a homotopy invariant but very hard to compute singular homology directly from the definition. One of the main results in the module will be the proof that simplicial homology and singular homology agree for simplicial complexes. This result means that we can combine the theoretical power of singular homology and the computability of simplicial homology to get many applications. These applications will include the Brouwer fixed point theorem, the Lefschetz fixed point theorem and applications to the study of vector fields on spheres.

Aims: To introduce homology groups for simplicial complexes; to extend these to the singular homology groups of topological spaces; to prove the topological and homotopy invariance of homology; to give applications to some classical topological problems.

Objectives: By the end of the module the student should be able to:

Give the definitions of simplicial complexes and their homology groups and a geometric understanding of what these groups measure
Use standard techniques for computing these groups
Give the extension to singular homology
Understand the theoretical power of singular homology
Develop a geometric understanding of how to use these groups in practice
The course is based on chapter 2 of Allen Hatcher's book:
Algebraic Topology, CUP. ( Available free from Hatcher's website).

Strongly recommended preliminary reading
Ideal for the summer holidays, and a good preparation also for MA3F1 Introduction to Topology:
David Richeson, Euler's Gem, Princeton, 2008
Jeffrey Weeks, The Shape of Space, Marcel Dekker, 2001

Additional references:
JW Vick, Homology Theory: an introduction to algebraic topology, Academic Press.
MA Armstrong, Basic Topology, Undergraduate Texts in Mathematics, Springer Verlag
Maunder, Algebraic Topology, Cambridge University Press.
A Dold, Lectures on Algebraic Topology, Springer-Verlag.



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Year 1 regs and modules
G100 G103 GL11 G1NC

Year 2 regs and modules
G100 G103 GL11 G1NC

Year 3 regs and modules
G100 G103

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