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 (email@example.com) no later than April 21 (2013).
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.