This course is taken in sequence, part 1 in the fall, and part 2 in the spring.

Prerequisites:

- Basic notions of differentiable manifolds (see Chapter 0 and 1 of do Carmo’s Riemannian Geometry).
- Fundamental group, van Kampen’s theorem, covering spaces.
- Commutative algebra of PIDs (free modules, classification of finitely generated modules).

### Algebraic Topology I

### FALL

Part I: Foundations of differential topology and homotopy theory

- Basic properties of higher homotopy groups
- Smooth manifolds, regular values and transversality
- Vector fields and the Poincaré-Hopf theorem
- Pontryagin construction
- Morse functions
- Topology of CW complexes
- Whitehead theorem

Part II: Homology and cohomology

- Homological algebra of chain complexes
- Singular and cellular homology
- The universal coefficient theorem
- Hurewicz theorem
- Cohomology and cup products
- Poincaré duality

### SPRING

Part I: Topology of fibrations

- Loopspaces
- Homology with local coefficients
- Obstruction theory
- Eilenberg MacLane spaces
- Filtered chain complexes and the associated spectral sequence
- The Serre spectral sequence of a fibration
- Multiplicative structures
- Applications to the computation of homotopy groups of spheres

Part II: Characteristic classes

- Classifying spaces of vector bundles
- Cohomology of Grassmannians
- Stiefel-Whitney, Chern, Pontryagin and Euler classes
- Thom spaces
- Computations of bordism groups
- Hirzebruch’s signature formula and Milnor’s exotic 7-spheres

Part III (time permitting): Further topics in Geometric Topology

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