Advanced Topology- continuation from Basic Topology:
πi(Sn); i ≤ n. [FF, 9], [HAT, 4.2].
CW complexes: de?nition, cellular approximation, CW aproximation, Whitehead theorem, computation of π1 and of homologies of CW complexes, obstacles to computation of πn of CW complexes. [GH, 21], [FF, 3], [HAT, 0, 4.1].
(a) Simplicial sets. De?nition, realisation, Kan condition. combinatorial description of homotopy classes of maps between realisations of Kan simplicial sets.
(b) long exact sequence of (Serre) ?bration. Examples. [FF, 7,8], [HAT, 4.2].
(c) Eilenberg-MacLane spaces [FF, 2], [HAT, 4.2].
(a) Relative homotopy groups and long exact sequence a pair. [FF, 8], [HAT, 4.1].
(b) Excision and corolaries: Hurewicz theorem, Freudenthal suspension theorem, stable homotopy groups [FF, 9], [HAT, 4.2].
4.4. Advanced Homology theory.
(a) Kunneth theorem. [GH, 29], [HAT, 3.2,3.B].
(b) Universal coe?cient theorem [GH, 29], [FF, 15], [HAT, 3.1, 3.A]
(c) Cohomology: de?nition, cup product, duality to homologies. [GH, 23, 24], [FF, 14], [HAT, 3.1].
(d) Cohomology with compact support and Borel-Moore homology. [GH, 26], [HAT, 3.3].
Cech (co-)homology. [HAT, 3.3].
(a) Orientation and Poincare duality [GH, 22, 26], [HAT, 3.3].
(b) Relation to Eilenberg-MacLane spaces [FF, 2], [HAT, 4.3]
4.5. Advanced topics.
(a) Sheaf cohomology.
(b) Spectral sequences.
(c) the stable homotopy category and spectra.
(d) Alexander duality
(e) Cohomology operations
(f) Bott periodicity theorem