Course on Homotopy Theory (first semester 2012/2013)

This is a course jointly taught by Moritz Groth and Ieke Moerdijk, and it is part of the Mastermath Program.

Time and place: Wednesday (first session: September 12), 10.15-13.00, room: Buys Ballot Lab, room BBL 023, Utrecht
Prerequisites: A course in point-set topology and one in group theory.
Exams: mid-term test-yourself-exam (to be solved at home and discussed in class, not graded) and a final written exam.
Retake: For thoses students who attended our Mastermath course on Homotopy Theory and failed the exam or were unable to take it, we offer the possibility of a second exam. This exam will be held on May 2, and its form (oral or written) will be decided based on the number of students registering for it. If you wish to take this exam, please send an e-mail to Greta Oliemeulen (g.oliemeulen@math.ru.nl) no later than April 21 (2013).

The course offers an introduction to algebraic topology centered around the theory of higher homotopy groups of a topological space. These groups offer more information than the homology or cohomology groups with which some students may be familiar, but are much harder to calculate. (In fact, the full computation of all the homotopy groups of spheres is unknown and in some sense is the 'holy grail' of algebraic topology, although many special cases are known.)

The course will start with a reminder about the fundamental group(oid), and the homotopy relation on maps. Next, we'll define the (higher) homotopy groups of a space, and prove some basic properties about them. We'll discuss the action of the fundamental group on these groups, and the Serre long exact sequence of a fibration. This will enable us to perform some elementary calculations.

We will discuss CW complexes and a proof of Whitehead's theorem about the construction of maps from the information about homotopy groups. Further, we will use CW complexes to construct the Postnikov tower of a space, a context in which the famous Eilenberg-MacLane spaces come up. The homotopy extensions and lifting property establishes an important relation between cofibrations and Serre fibrations (this is the motivation for one of Quillen's axioms for 'homotopipcal algebra', axioms which play a dominant role in much of modern algebraic topology). We conclude the course by the Homotopy excision theorem and the Freudenthal suspension theorem, key results that lies at the basis of stable homotopy theory.

Lecture notes:
  Section 01   Homotopy and the fundamental groupoid
  Section 02   The compact-open topology and loop spaces
  Section 03   Higher homotopy groups
  Section 04   Relative homotopy groups and the action of the fundamental group
  Section 05   Fibrations and homotopy fibers
  Section 06   Fiber bundles
  Section 07   CW complexes and basic constructions
  Section 08   Cofibrations
  Section 09   Cellular approximation
  Section 10   CW approximation and Whitehead's theorem
  Section 11   Killing homotopy groups: Postnikov and Whitehead towers
  Section 12   Homotopy extension and lifting property
  Section 13   Homotopy excision and the Freudenthal suspension theorem

Exercise sheets:
  Exercise sheet 01
  Exercise sheet 02
  Exercise sheet 03
  Exercise sheet 04
  Exercise sheet 05
  Exercise sheet 06
  Exercise sheet 07
  Exercise sheet 08
  Exercise sheet 09 (This is the test-yourself-exam.)
  Exercise sheet 10
  Exercise sheet 11
  Exercise sheet 12

References: There are many references for such a course including the following ones:
  Switzer: Algebraic Topology: homotopy and homology (first part)
  Davis, Kirk: Lecture Notes in Algebraic Topology (Chapter 6 and 7)
  Arkowitz: Introduction to Homotopy Theory

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