Zoran Skoda homCW

Zoran Škoda: Homotopy of CW-complexes

(advanced graduate course, 60 hours, 2014/2015)

Prerequisites

Basic general topology, basic familiarity with notions of fundamental group, covering space, manifold, chain complex, being comfortable with notions of a functor, equivalence of categories and natural transformation, undergraduate algebra. It is not necessary to have taken previously graduate course Geometry and topology as the overlap is small and the definitions will be repeated.

Description

This is an introduction to the homotopy theory of topological spaces with the emphasis on the weak homotopy type and CW-complexes. The choice of topics is similar to Switzer’s monograph though the volume of material is smaller and some complementary topics added. Focus on nice spaces allow simpler and quicker building of the theory, so that more advanced, but standard, objects, (and tools of modern homotopy theory), including spectra, could be approached.

• Short summary of needed general topology. Exponential law and compactly generated spaces. Basic language of category theory. Based spaces and pairs.
• (General notions of homotopy theory) Serre and Hurewitz fibrations, cofibrations, homotopy sets and groups and weak equivalences; mapping cone and cylinder, path space, suspension,
  loop space, free loop space, smash product, join, H-monoids, H-(co)groups.
• CW-complexes, CW-approximation, Whitehead’s theorem, cellular (co)homology, axioms of generalized (co)homology theories.
• Freudenthal suspension teorem, stable homotopy groups, $\Omega$-spectra, Brown representability theorem, $K(\pi,n)$-spaces, fibre bundles, classifying space $BG$ and universal bundle $EG\to BG$.
• Main examples of generalized (co)homology theories including K-theory and cobordism. Orientation, dualities, products and cohomology operations.
• Killing and cokilling homotopy groups. Postnikov decomposition. Whitehead tower. Obstruction theory and characteristic classes.

Literature

1. R. Switzer, Algebraic topology: homology and homotopy, Springer 1975

2. P. May, Concise introduction to algebraic topology, Revised version,

http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf

Most material will be prepared from Switzer and sometimes May; however in the preparation of individual lectures I may consult approaches of other standard textbooks (in particular Spanier, Whitehead and Postnikov).