Math 701- Homotopy Theory

IMPORTANT:

!The classes will be face-to-face!
!Attendance of lectures is required.!
!If you want to follow the course as a guest student, please send me an email: pasemra@metu.edu.tr!

Course Code: 2360701

Schedule: Mondays: 13:40-15:30
                     Wednesdays: 13:40-15:30

Content: Homotopy groups, Whitehead's theorem, CW approximation; homotopy excision, Hurewicz theorem; (co)fibrations, mapping path and loop spaces; fibre bundles, sphere bundles over spheres; obstruction theory, relation to cohomology; Postnikov towers.

Course Objectives:
This course provides students with a solid working knowledge in the basic techniques of Homotopy Theory and constitutes a natural continuation of the Math 537-538 sequences in Algebraic Topology. Topics will center around properties and calculations with higher homotopy groups as well as the more general theory of fibrations and fiber bundles. The course should be of interest to all students with research interests in topology or geometry.

Course Learning Outcomes:
By the end of the course, the student must be able to: 

• Manipulate fibrations and cofibrations.
• Perform elementary computations of homotopy groups.
• Compare homotopy with homology groups.
• Define the notions introduced in the course.
• State the main theorems and prove them.
• Apply the tools developed in the course to examples.
  

Tentative Weekly Outline:

1-Review of CW complexes, higher homotopy groups
2-Relative homotopy groups, functorial properties, fundamental group actions
3-Pair exact sequence, compression lemma , Whitehead Theorem
4-Cellular and CW approximation
5-Whitehead & Postnikov Towers
6-Homotopy Excision Theorem and computations
7-Moore spaces and Eilenberg-MacLane space
8-Hurewicz maps, general Hurewicz Theorem
9-Homotopy Lifting Property, (Serre) Fibrations, fiber bundles, cofibrations
10-Long Exact Sequences for fibrations, applications to spheres and Lie groups
11-Whitehead products, stable homotopy groups, ring structures
12-Loop spaces & Suspension, exact and coexact Puppe sequences
13- Relations to cohomology theory and characteristic classes
14-Obstruction Theory


Course Textbook(s):

1. Allen Hatcher, “Algebraic Topology”. (Chapter 4) Cambridge Univ Press, 2002.
2. Davis, J. F. and Kirk, P.,  "Lecture notes in algebraic topology" (Chapter 4,6,7) (Graduate Studies in Mathematics, no. 35, American Mathematical Society, 2001)

Supplementary Readings :

1. Brayton Gray: "Homotopy theory; an introduction to algebraic topology", Academic Press Series of Monographs and Texts (# 64).
2. G. W. Whitehead: "Elements of Homotopy Theory". GTM 61, Springer 1978

Assessment of Student Learning/ Grading::
Assignments %70: Homework assignments will be given during the course.
Final Exam %30: Presentation given by student about some topic given by the lecturer at the end of term.