Algebraic Topology

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Description

Thursday 13.00-15.00

Module Overview: Algebraic Topology

This module is intended to give an overview of basic concepts, examples and techniques in algebraic topology. 

Lectures 

Provisional programme:

  • Lecture 1: Basic examples and constructions of topological spaces.
  • Lecture 2: Manifolds, basic homotopy theory and homotopy groups.
  • Lecture 3: Cofibrations, cell attachments and CW-complexes.
  • Lecture 4: Cellular approximation and relative homotopy groups.
  • Lecture 5: Fibre bundles, fibrations and the Hopf map.
  • Lecture 6: An introduction to homology.
  • Lecture 7: Homotopy invariance, exactness and excision.
  • Lecture 8: Computations and applications of homology.
  • Lecture 9: An introduction to cohomology
  • Lecture 10: Further topics in cohomology theory

Assessment 

This module is assessed in two assignments, each equally weighted to be worth 50% of the Final Grade. 

Assignment 1: will be released after the lecture in week 5.

Assignment 2: will be released at the end of the module.

 

Recommended references

  • Algebraic Topology  by A. Hatcher : An excellent book, covering all the material of the course (and much more) in great detail. Geometrically-minded and with many good pictures, it always tries to build the geometric intuition before jumping into the abstract algebra. He tends to write long explanations: it's excellent to read for learning the topic, but can be impractical when searching for specific results. Freely available on the author's website https://pi.math.cornell.edu/~hatcher/
  • A Concise Course in Algebraic Topology  by J. May : Another good book covering all we need. As the title suggests, it is of somewhat opposite style compared to Hatcher. He has a more algebraic/categorical approach, no figures but many commutative diagrams. Being more concise, it can be hard as a first introduction to the subject. Excellent as a complement to Hatcher, and for those algebraically minded!
  • Topology and Geometry  by G. E. Bredon : Another classical reference with a rather geometric viewpoint. Compared to Hatcher, it sometimes lacks details, which can make a first reading a bit difficult. On the other hand, it includes some background of point-set topology and material about differential topology (which we will not discuss in the course but could be interesting to some of you).