# Part III - Algebraic Topology

## Lectured by O. Randal-Williams, Michaelmas 2016

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# Contents

- V Full version
- 1 Homotopy
- 2 Singular (co)homology
- 3 Four major tools of (co)homology
- 3.1 Homotopy invariance
- 3.2 Mayer-Vietoris
- 3.3 Relative homology
- 3.4 Excision theorem
- 3.5 Applications
- 3.6 Repaying the technical debt
- 4 Reduced homology
- 5 Cell complexes
- 6 (Co)homology with coefficients
- 7 Euler characteristic
- 8 Cup product
- 9 Kunneth theorem and universal coefficients theorem
- 10 Vector bundles
- 10.1 Vector bundles
- 10.2 Vector bundle orientations
- 10.3 The Thom isomorphism theorem
- 10.4 Gysin sequence
- 11 Manifolds and Poincare duality