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.

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.

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 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

Switzer: Algebraic Topology: homotopy and homology (first part)

Davis, Kirk: Lecture Notes in Algebraic Topology (Chapter 6 and 7)

Arkowitz: Introduction to Homotopy Theory