email: debasis@iitk.ac.in
Office no FB 501,
Department of Mathematics and Statistics,
IIT Kanpur.
Course overview:
Homotopy theory is a branch of topology that studies spaces up to continuous deformation. The aim of the course is to introduce the basic techniques in homotopy theory. These tools are ubiquitous throughout modern mathematics. We will study some properties of the homotopy category of spaces, and discuss some computational tools.
Algebraic Topology : A.Hatcher
Concise Algebraic Topology : Peter May
Geometry and Topology : G. Bredon
Survey article by A. Noalekar (ISI Banglore)
1 CW complexes, higher homotopy groups
(Ref: done in class)
2 Relative homotopy groups, properties, action of fundamental group.
(Ref: done in class)
3 Long exact sequence of a pair, Whitehead Theorem.
(Ref: done in class)
4 Cellular approximation and CW approximation.
(Ref: done in class)
5 Postnikov Towers, k-invariants, Whitehead towers.
(Ref: Hatcher chapter 4 page 410, proposition 4.21 and 4.22, chapter 22.4 May's book, Note1, Note2)
6 Homotopy Excision Theorem/Blakers-Massey Theorem, Freudenthal Suspension Theorem.
(Ref: Chapter 11 May's book, Section 4.2 of Hatcher book, Note1, Note2)
7 Moore spaces (Hatcher Example 2.40, example 4.34 page 368) and Eilenberg–MacLane space (Hatcher Section 4.2, page 365. Note1).
8 Hurewicz Theorem.
(Ref: Hatcher Theorem 4.32, Peter May: Chapter 15.1, Note1 Theorem 3.1)
9 Homotopy Lifting Property, Fibrations, fiber bundles.
(Ref: done in class, read Fiber bundle from Hatcher section 4.2, page 365)
10 Long Exact Sequences for fibrations, applications to spheres.
(Ref: May's book chapter 9.3, 9.4, Hatcher example 4.49 to example 4.55)
11 Whitehead products (Hatcher Example 4.52), stable homotopy groups, ring structures (Hatcher page 384. Proposition 4.56).
12 Loop spaces & Suspension (Ref: done in class), exact and co-exact Puppe sequences. (Ref: May fiber and cofiber sequences 4.4 and 4.6, )
13 Relations to cohomology theory.
(Ref: May's book section 22.2, note1)
14 Obstruction Theory.
(Ref. Hatcher section 4.3 page 415, note1)
Exercises: