Syllabus (preliminary!)
-- On history of algebraic topology.
-- Basic objects and categories: manifolds, simplicial complexes, cell complexes. -- Basic examples. -- Topological manifolds versus smooth manifolds. -- Every smooth manifold homotopic to a cell complex: Morse theory.
-- Classification of 2-dimensional manifolds.
-- Examples of 3-dimensional and 4-dimensional manifolds.
== Homology and cohomology ==
-- Simplicial and singular homology and cohomology. Betti numbers.
-- Axioms for the homology and cohomology theories.
-- Homological algebra - 1: resolutions, homotopic complexes, quasi-isomorphisms.
-- De Rham cohomology of smooth manifolds.
-- Sheaf (Cech) cohomology of the constant sheaf.
-- Homological algebra - 2: Derived functor of a left exact functor. Long exact sequence associated to a short exact sequence.
-- Isomomorphisms: idea of the proof.
-- (If time permits): Cohomology ring. -- Poincare duality for manifolds. -- Pairing in the middle cohomology of manifolds.
= Homotopy =
-- Fundamental group. -- Examples of computations. Covering spaces, and topological Galois theory. Example: solving polynomials of degree 5.
-- (If time permits): Higher homotopy groups. Eilenberg-Maclane spaces K(G,n). -- Postnikov tower.
= If we get lucky; or: some time in the future =
Fundamental class of a submanifold in de Rham cohomology. -- Thom isomorphism, and applications. Also: Vector bundles. -- Characteristic classes via obstruction theory, and via maps to classifying spaces. -- K-groups. -- Cohomology of some classifying spaces. -- Relation to the cohomology of Lie groups.