This is a course on basic aspects of algebraic topology offered by Moritz Groth
and Ieke Moerdijk. The course consists of roughly *twelve to fourteen lectures*.

**Time and place:** Monday (first session: September 9), 10.45-12.30, **the room varies a lot**, namely HG01.029 (weeks 37-39), HG00.310 (weeks 40-41), HG03.054 (week 42), HG00.310 (week 43), HG00.633 (weeks 44-47), HG00.068 (weeks 48-49), HG00.633 (weeks 50-51)

**Prerequisites:** Basic point set topology and algebra.

**Examination:** Grade based on presentations by students and oral exam.

This course offers an introduction to algebraic topology, i.e, the study
of topological spaces by means of algebra. The first part of the course
focuses on homology theory. Singular homology groups are algebraic
invariants of spaces: for every space there are such groups and every map
of spaces induces a map between the corresponding groups. These invariants
turn out to be rather computable, and they allow for some immediate
geometric applications.
We will establish some key properties of these homology groups like the
homotopy-invariance and the excision theorem. A convenient variant is
provided by singular homology with coefficients - a framework which makes
necessary a short discussion of basic homological algebra including tensor
and torsion products.
For CW complexes, there is also the more combinatorial cellular homology
theory. The course culminates in a proof that singular homology and
cellular homology agree on CW complexes. This allows for more explicit
calculations in examples of interest (e.g. projective spaces).
In the sequel Algebraic Topology II
we discuss a bit more advanced subjects like cohomology theory.

Lecture 01 Definition of singular homology

Lecture 02 Low-dimensional identifications

Lecture 03 Relative singular homology

Lecture 04 Singular homology of contractible spaces

Lecture 05 Homotopy invariance of singular homology

Lecture 06 Excision property and Mayer-Vietoris sequence

Lecture 07 Proof of excision property of singular homology

Lecture 08 Degree of a map

Lecture 09 CW complexes

Lecture 10 Basic homological aspects of CW complexes

Lecture 11 Cellular homology

Lecture 12 Isomorphism between cellular and singular homology

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

Exercise sheet 10

Exercise sheet 11

Exercise sheet 12

Classification of covering spaces (Sandra Hommersom, Merlyn Keune, January 20, 15.00-15.45, HG03.085)

Homology with coefficients and universal coefficients theorem (Jorma Dooper, January 20, 16.00-16.45, HG03.085)

Simplicial complexes and simplicial homology (Dennis Hendrikx, Laura van den Berge, January 21, 15.00-15.45, HG03.085)

Classification of surfaces and their homology (Daan van der Maal, Niels van der Weide, January 21, 16.00-16.45, HG03.085)

Laura van den Berge, January 22, 16.00-16.45)

Daan van der Maal, January 22, 17.00-17.45)

Dennis Hendrikx, January 23, 16.00-16.45)

Jorma Dooper, January 23, 17.00-17.45)

Merlyn Keune, January 24, 10.15-11.00)

Niels van der Weide, January 24, 11.15-12.00)

Sandra Hommersom, January 24, 12.15-13.00)