# Course on Algebraic Topology I (first semester 2013/2014)

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 notes:
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 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
Exercise sheet 10
Exercise sheet 11
Exercise sheet 12

Projects (presented by students):
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)

Schedule for the orals:
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)

References: will soon appear here.