We will discuss homotopy and homology theory. The course is split into two units. The first one contains the most elementary facts of those theories, together with their detailed proofs. The second will contains more advanced material of both theories, The last lecture(s) will be an overview of more advance topics The course is of M.Sc. level. It includes some basic geometry facts every mathematician is expected to know. Math students are strongly recommended to attend, CS or physics students wishing to broaden their mathematical background are also welcome.
- Motivation and overview.
- Basic Homotopy theory: homotopy, homotopy category, homotopy equivalence, pointed topological space. [GH, 1,2], [FF, 1], [HAT, 0].
Operation with spaces: product, bouquet, quotient, smash product, suspension, join, loop space,(mapping) cylinder and (mapping) cone. [GH, 7], [FF, 1], [HAT, 0]
fundamental group π1: de?nition, homotopy invariance, coverings, universal covering (existence and uniqueness), relation between coverings and π1, examples. [GH, 4-6], [FF, 4,5], [HAT, 1.1,1.3].
π1 of a bouquet product and suspension, Seifert-van Kampen theorem, equivalent de?nitions of π1, fundamental groupoid. [HAT, 1.2].
Higher homotopy groups πn (basic facts): de?nition, commutativity, homotopical groups and co-groups. πn of products, coverings and loop spaces, di?culties of computation of πn of bouquets and suspensions. Weak homotopy equivalence of topological spaces, examples. [GH, 7], [FF, 6], [HAT, 4.1].
Simplicial complexes: de?nition, realization. Whitehead theorem: weak homotopy equivalence of simplicial complexes implies their homotopy equivalence. Barycentric subdivision. Any topological space is weak homotopy equivalent to a simplicial complex. [HAT, 2.1, 4.1].
(a) Euler theorem, Euler characteristic of a simplicial complex.
(b) Homologies of a simplicial complex: de?nitions, examples. [GH, 10], [HAT, 2.1].
Axiomatic approach to homologies: De?nition, Barratt-Puppe sequence, relative homologies. Some corollaries and equivalent axioms: Mayer-Vietoris theorem, excision theorem, Hn of bouquet, long exact sequence of a triple, examples, uniqueness, Generalized Homology theories, problems with Hn of loop space. [GH, 16-17], [FF, 12], [HAT, 2.2, 2.3].
Singular homologies: de?nition, proof of axioms. [GH, 14-15], [FF, 11], [HAT, 2.1].
4.3. Advance Homotopy theory.