Algebraic topology 1000-OG-EN-TA

Lecture:

1 Topological spaces, continuous maps, subspaces, products, quotient spaces.
2 Connectedness, path-connectedness, compactness; basic examples and constructions.
3 Review of groups, rings, modules, homomorphisms, quotients, and exact sequences. Categories, functors, natural transformations, commutative diagrams.
4 Paths, homotopies of paths, homotopies of maps, homotopy equivalence.
5 The fundamental group: definition, functoriality, change of base point; examples.
6 Computation of fundamental groups; Seifert–van Kampen theorem and its applications.
7 Covering spaces; examples; lifting of paths and homotopies.
8 Universal coverings and applications to fundamental groups.
9 Higher homotopy groups; basic properties, functoriality, and homotopy invariance.
10 Singular chains, boundary operators, singular homology groups; homology of a point and simple spaces.
11 Relative homology; long exact sequence of a pair; excision.
12 Mayer–Vietoris sequence and computations of homology groups.
13 Homotopy invariance of homology; axiomatic characterization and uniqueness of homology theory.
14 Euler characteristic and its relation to homology. Homological properties of spheres and disks.
15 Application: Brouwer fixed-point theorem.

Exercises: Calculating homotopy and homology groups in examples; completing details of some proofs arranged as independent problems.