Algebraic Topology, WS 2021/22
Prof. Dr. Clara Löh
/
Matthias Uschold
/
Johannes Witzig
News

Registrations for exams:
The exams will be oral exams (25 mins) on zoom. Dates: 18.02., 11.03., 04.04.
An additional option in March is be offered (11.03.).
Algorithm:

Register with Ms Bonn for a date/timeslot for an exam (rosina.bonn@ur.de, +49/941/9432766).

Register in FlexNow for the corresponding date
In case you don't know what to expect from such an exam:
The
Fachschaft
has a collection of minutes of exams from students.
Technical prerequisites: You will need some device to share
handwritten notes that you write during the exam, e.g., a graphics
tablet, an ordinary tablet, a document camera, a mobilephoneusedasdocumentcamera, ...
There are many options!

Registration for the Studienleistung: via FlewNow;
please register before the deadline.

Followup courses in SS 2022:

10.01.2022:
Until further notice:

We will continue the lectures in the hybrid format (i.e.,
in person lectures on campus with a live zoom stream).

The exercise class of Matthias will be held in hybrid
format.

The exercise class of Johannes will be held in person;
participants that want/need to attend online should join the
zoom stream of the exercise class of Matthias.

Exercise sheet 15 (of February 8) is online: pdf;
bonus sheet; optional submission before February 15, 8:30 (via GRIPS)

The lecture notes are updated:
pdf
(version of 11.02.2022)

The results of the course evaluation can be found on GRIPS; thanks
for all your feedback!

Organisational matters

The time/location changed! (In order to resolve the
time conflict with Differential Geometry I.)

If you plan to write a bachelor thesis under my supervision in SS 2022 (in
Topology/Geometry), you should participate in a seminar in the Global Analysis
and Geometry group before SS 2022.
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,
(non)embeddability results, topological data analysis, and many
more. 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 "Geometric Group Theory" in the summer 2022.
The course in SS 2022 can also be attended independently of the present course on Algebraic Topology.
Moreover, there probably will also be a continuation of the Algebraic Topology Series.
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:
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
Natural Transformations]

Homotopy and Homotopy Invariance
[Homotopy,
Homotopy Invariance,
Using Homotopy Invariant Functors]

Fundamental Group and Covering Theory

The Fundamental Group
[The Group Structure on the Fundamental Group,
Changing the Basepoint]

Divide and Conquer
[The Fundamental Example,
Products,
Glueings,
Computations,
Ascending Unions]

Covering Theory
[Coverings,
Lifting Properties,
The Universal Covering,
The Fundamental Example,
The Classification Theorem,
Application: The NielsenSchreier Theorem
]

Axiomatic Homology Theory

The EilenbergSteenrod Axioms
[The Axioms,
First Steps]

Homology of Spheres and Suspensions
[Suspensions,
Homology of Spheres,
Mapping Degrees of SelfMaps of Spheres]

Glueings: The MayerVietoris sequence

Classification of Homology Theories

Singular Homology

Construction
[Geometric Idea,
Singular Homology,
First Steps,
The Long Exact Sequence of Pairs
]

Homotopy Invariance
[Geometric Idea,
Prism Decomposition,
Proving Homotopy Invariance]

Excision
[Geometric Idea,
Barycentric Subdivision,
Proving Excision]

Applications
[Singular Homology as Ordinary Homology,
The Jordan Curve Theorem,
Invariance of Domain and NonEmbeddability,
(Commutative Division Algebras),
(Rigidity in Geometry)]

Singular Homology and Homotopy Groups

Celluar Homology

CWComplexes

Cellular Homology
[Geometric Idea,
The Construction,
Comparison of Homology Theories]

The Euler Characteristic
[Geometric Definition of the Euler Characteristic,
A Homological Description,
Divide and Conquer,
Application: NielsenSchreier, Quantitatively]

Appendix

PointSet Topology

Homotopy Flipbooks:
one example: pdf
and another example: pdf

Cogroup Objects and Group Structures

Amalgamated Free Products

Group Actions

Basic Homological Algebra
[Exact Sequences,
Chain Complexes and Homology,
Homotopy Invariance]

Homotopy Theory of CWComplexes
[Whitehead Theorem,
Cellular Approximation,
Subcomplexes and Cofibrations]
Notes from online teaching:

Current lecture:
pdf

14.12.2021:
pdf

17.12.2021
pdf

21.12.2021
pdf

07.01.2022
pdf
Time/Location
Tuesday, 8:3010:00, M 101,
Friday, 8:3010:00, M 101.
Exercise classes
Time/Location:
Fr 1012, M009, Matthias Uschold
Fr 1214, M103, Johannes Witzig

Please register before October 20, 10:00, via GRIPS for the exercise classes!

The exercise classes start in the second week.

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 19, 2021,

submission before October 26, 2021 (08:30)

will be discussed in the exercise classes in the second week

Sheet 2,

of October 26, 2021,

submission before November 2, 2021 (08:30)

will be discussed in the exercise classes on 05.11.2021




Projective Asteroids
(proof of concept, by Johannes Witzig)

Sheet 3,

of November 2, 2021,

submission before November 9, 2021 (08:30)

will be discussed in the exercise classes on 12.11.2021

Sheet 4,

of November 9, 2021,

submission before November 16, 2021 (08:30)

will be discussed in the exercise classes on 19.11.2021

Sheet 5,

of November 16, 2021,

submission before November 23, 2021 (08:30)

will be discussed in the exercise classes on 26.11.2021

Sheet 6,

of November 23, 2021,

submission before November 30, 2021 (08:30)

will be discussed in the exercise classes on 03.12.2021

Sheet 7,

of November 30, 2021,

submission before December 7, 2021 (08:30)

will be discussed in the exercise classes on 10.12.2021

Sheet 8,

of December 7, 2021,

submission before December 14, 2021 (08:30)

will be discussed in the exercise classes on 17.12.2021

Sheet 9,

of December 14, 2021,

submission before December 21, 2021 (08:30)

will be discussed in the exercise classes on 07.01.2022

Sheet 10,

of December 21, 2021,

submission before January 11, 2022 (08:30)

will be discussed in the exercise classes on 14.01.2022

Sheet 11,

of January 11, 2022,

submission before January 18, 2022 (08:30)

will be discussed in the exercise classes on 21.01.2022

Sheet 12,

of January 18, 2022,

submission before January 25, 2022 (08:30)

will be discussed in the exercise classes on 28.01.2022

Sheet 13,

of January 25, 2022,

submission before February 1, 2022 (08:30)

will be discussed in the exercise classes on 04.02.2022

Sheet 14,

of February 1, 2022,

submission before February 8, 2022 (08:30)

will be discussed in the exercise classes on 11.02.2022

Sheet 15,

of February 8, 2022,

bonus sheet; optional submission before February 15, 2022 (08:30)


Études
These études 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 19 2021,

no submission

Sheet 1,

of October 22 2021,

no submission

Sheet 2,

of October 29 2021,

no submission

Sheet 3,

of November 5 2021,

no submission

Sheet 4,

of November 12 2021,

no submission

Sheet 5,

of November 19 2021,

no submission

Sheet 6,

of November 26 2021,

no submission

Sheet 7,

of December 3 2021,

no submission

Sheet 8,

of December 10 2021,

no submission

Sheet 9,

of December 17 2021,

no submission

Sheet 10,

of January 7, 2022,

no submission

Sheet 11,

of January 14, 2022,

no submission

Sheet 12,

of January 21, 2022,

no submission

Sheet 13,

of January 28, 2022,

no submission

Sheet 14,

of February 4, 2022,

no submission

Sheet 15,

of February 11, 2022,

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.
Exams
Please read the
information on organisation and formalities of this course.
Last change: February 11, 2022