Topology

Aims and objectives for the course

Objectives: The student should

1. Become familiar with the basic language of topological spaces, and be able to in principle make rigorous all arguments presented using pictures (identification spaces and so on).
2. See how the notion of homotopy equivalence leads to the fundamental group, and its significance for detecting holes in a space.
3. Understand how the fibre theorem and covering spaces allow fundamental groups to be computed, and understand how knowledge of fundamental groups can prove deep fixed-point and other theorems.
4. Understand both theoretically and as an algorithm the standard form for and resulting classification of closed surfaces.
5. Be able to compute directly the homology groups of a low- dimensional simplicial complex, and understand how the ranks of the homology groups are related to information about higher-dimensional `holes' in the space.
6. Be able to compute homology groups of low- dimensional spaces using appropriate triangulations, and to be able to state the usual invariance and well-definedness properties of homology groups.
7. To understand how higher homology groups may be used (assuming the standard results on simplicial approximation) to prove Brouwer fixed point theorem.
8. To be able to describe without proofs and use the long exact sequence of a pair. This requires an understanding of the maps involved, and the ability to do simple calculations in exact sequences of groups.

Lecture notes in pdf

Homework Sheets