Algebraic Topology, WS 2018/19
Prof. Dr. C. Löh
/
D. Fauser
/
J. Witzig
News

The lecture notes are updated (version of October 22).

The new exercise sheet is online: Sheet 2
(sheet of October 22, submission before October 29, 10:00).
Solutions can be submitted in English or German and in teams of up to two people.
Please do not forget to add your name to all your submissions!

The new etudes are online: Etudes 1
(sheet of October 19, no submission).
These etudes help to train elementary techniques and
terminology. These problems should ideally be easy enough to be solved
within a few minutes. Solutions are not to be submitted and will not
be graded.

Please register before October 17, 10:00, via GRIPS for the exercise classes!

Organisational matters

If you are interested in taking this course in WS 2018/2019, please register as a course participant in HIS/LSF
(if possible, before July 9), so that the number of teaching assistants can be determined. In case you
do not have yet an RZaccount, you can also send an email to clara.loeh@mathematik.unir.de to register.

The seminar
Topology vs. Combinatorics
in WS 2017/18 accompanies this course; no previous knowledge of algebraic topology is required.
The organisational meeting is on Friday, July 6, 13:45, M 201.

If you plan to write a bachelor thesis under my supervision in SS 2019 (in
Topology/Geometry), you should participate in a seminar in the Global Analysis
and Geometry group before SS 2019.
Algebraic Topology
Algebraic topology studies topological spaces via algebraic invariants  by modelling certain
aspects of topological spaces in the realm of algebra, e.g., by groups and group homomorphisms.
Classical examples include homotopy groups and (co)homology theories.
Algebraic topology has various applications, both in theoretical and in applied mathematics,
for instance, through fixed point theorems and (non)embeddability results. For example,
Nash's proof of existence of certain equilibria in game theory is based on a topological
argument. Topics covered in this course include:

What is algebraic topology?

The fundamental group and covering theory

The EilenbergSteenrod axioms

Singular homology

Cellular homology

Classical applications of (co)homology.
This course will be complemented with the course "Group Cohomology" in the summer 2019, where (co)homology of groups will be studied. The course in SS 2019 can also be attended independently of the present course on Algebraic Topology.
If all participants agree, this course can be held in German; solutions to the
exercises can be handed in in German or English.
Lecture notes:
pdf.
Topics covered so far:

Literature
 Introduction

What is Algebraic Topology?

Topological Building Blocks
[Construction: Subspaces,
Construction: Products,
Construction: Quotient Spaces and Glueings,
The Homeomorphism Problem]

Categories and Functors
[Categories,
Functors]

Appendix
Time/Location
Monday, 1012, M 102,
Thursday, 1012, M 104.
Exercise classes
Do 1416, H31
Fr 810, M009

The exercise classes start in the first week; in this first session, some
basics on topological spaces and categories will be discussed (as on the
sheet Etudes 0).

Organisational matters
Exercise sheets
Solutions can be submitted in English or German and in teams of up to two people.
Please do not forget to add your name to all your submissions!
Sheet 1,

of October 15, 2018,

submission before October 22, 2018 (10:00)

will be discussed in the exercise classes on October 25/26

Sheet 2,

of October 22, 2018,

submission before October 29, 2018 (10:00)

will be discussed in the exercise classes on November 2/?

Etudes
These etudes help to train elementary techniques and
terminology. These problems should ideally be easy enough to be solved
within a few minutes. Solutions are not to be submitted and will not
be graded.
Sheet 0,

of October 15, 2018,

no submission,

will be discussed in the exercise classes on October 18/19.

Sheet 1,

of October 19, 2018,

no submission,


Literature
This course will not follow a single book. Therefore, you should
individually compose your own favourite selection of books.
A list of suitable books can be found in the lecture notes.
Prerequisites
All participants should have a firm background in Analysis I/II
(in particular, basic point set topology, e.g., as in
Analysis II in WS 2011/12),
in Linear Algebra I/II, and basic knowledge in group theory
(as covered in the lectures on Algebra).
Knowledge about manifolds as in Analysis IV is not necessary, but helpful.
Knowledge about basic homological algebra (as in the last two weeks
of
Kommutative Algebra
is not necessary, but helpful.
Last change, October 22, 2018