Algebraic Topology I, WS 2015/16
Prof. Dr. C. Löh
News

The final exercise sheet (no submission; submitted solutions will
count as bonus points) is online

The pdf summary is updated (version of February 5).

Please register in FlexNow for the exam (and the Studienleistung, if applicable). Moreover, for the
oral exam, you also need to register with Ms. Bonn (M 217) for a time slot. (Details: see bottom of the page)

This course will be continued in the summer semester 2016 with the course Algebraic Topology II (by Dr. George Raptis). Details will be announced soon.

Moreover, the seminar
Coxeter Groups
in SS 2016 provides an opportunity
to apply and extend the skills from the course Algebraic Topology I.
The organisational meeting was on Wednesday, January 27, 9:15, M 101. There still are
free slots! and you can register by email.

The teaching evaluation of this course was on November 27, 11:00. The results
are available in GRIPS.

There is a seminar complementing the contents of this course:
Seminar: Simplicial Topology

Please read the information on organisation and formalities of this course.

If you plan to write a bachelor thesis under my supervision in SS 2016 (in
Topology/Geometry), you should participate in a seminar in the Global Analysis
and Geometry group before SS 2016.
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 continued in the summer 2016 with the course "Algebraic Topology II".
If all participants agree, this course can be held in German; solutions to the
exercises can be handed in in German or English.
Summary (in German):
pdf.
Topics covered so far:

Literaturhinweise
 Einführung

Was ist algebraische Topologie?
[Topologische Bausteine,
Kategorien und Funktoren,
Homotopie,
Homotopieinvarianz]

Fundamentalgruppe und Überlagerungstheorie
[Eine Gruppenstruktur auf pi1,
Die Fundamentalgruppe  divide et impera,
Überlagerungen  Grundbegriffe,
Überlagerungen  Liftungseigenschaften,
Überlagerungen  Klassifikation,
Anwendungen]

Axiomatische Homologie
[Die EilenbergSteenrodAxiome,
Fingerübungen,
Homologie von Sphären und Einhängungen,
Die MayerVietorisSequenz,
Ausblick: Existenz und Eindeutigkeit von Homologietheorien
]

Singuläre Homologie
[Anschauliche Skizze der Konstruktion,
Konstruktion singulärer Homologie,
Homotopieinvarianz von singulärer Homologie,
Ausschneidung in singulärer Homologie,
singuläre Homologie als gewöhnliche Homologietheorie,
Homologie, Singuläre Homologie und Homotopiegruppen]

Zelluläre Homologie
[CWKomplexe, zelluläre Homologie,
Vergleich von Homologietheorien auf CWKomplexen,
Die EulerCharakteristik]

Anhang: Grundbegriffe aus der mengentheoretischen Topologie
[Topologische Räume,
stetige Abbildungen,
(Weg)Zusammenhang,
Hausdorffräume,
Kompaktheit]

Anhang: Kogruppenobjekte und induzierte Gruppenstrukturen

Anhang: Freie (amalgamierte) Produkte von Gruppen
[Freie (amalgamierte) Produkte,
freie Gruppen]

Anhang: Gruppenoperationen

Anhang: Homologische Algebra
[exakte Sequenzen,
Kettenkomplexe und Homologie,
Kettenhomotopie
]

Anhang: Wörterbuch
Time/Location
Tuesday, 1012, M 104,
Friday, 1012, M 104.
Exercise classes
Wed, 1618, M102. First exercise class on Oct 14.
The exercise class will be held by
Dr. Dmitri Pavlov and Gerrit Herrmann.
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 13, 2015,

no submission,

will be discussed in the exercise class on October 14.

Sheet 0,

of October 16, 2015,

no submission,

will be discussed in the exercise class on October 21.

Sheet 1,

of October 16, 2015,

submission by October 23,

will be discussed in the exercise class on October 28.

Sheet 2,

of October 23, 2015,

submission by October 30,

will be discussed in the exercise class on November 4.

Sheet 3,

of October 30, 2015,

submission by November 6,

will be discussed in the exercise class on November 11.

Sheet 4,

of November 6, 2015,

submission by November 13,

will be discussed in the exercise class on November 18.

Sheet 5,

of November 13, 2015,

submission by November 20,

will be discussed in the exercise class on November 25.

Sheet 6,

of November 20, 2015,

submission by November 27,

will be discussed in the exercise class on December 2.

Sheet 7,

of November 27, 2015,

submission by December 4,

will be discussed in the exercise class on December 9.

Sheet 8,

of December 4, 2015,

submission by December 11,

will be discussed in the exercise class on December 16.

Sheet 9,

of December 11, 2015,

submission by December 18,

will be discussed in the exercise class on January 13.

Sheet 10,

of December 18, 2015,

submission by January 8,

will be discussed in the exercise class on January 13.

Sheet 11,

of January 8, 2016,

submission by January 15,

will be discussed in the exercise class on January 20.

Sheet 12,

of January 15, 2016,

submission by January 22,

will be discussed in the exercise class on January 27.

Sheet 13,

of January 22, 2016,

submission by January 29,

will be discussed in the exercise class on February 3.

Sheet 14,

of January 29, 2016,

no submission


Sheet 15,

of February 5, 2016,

no submission


Literature
This course will not follow a single book. Therefore, you should
individually compose your own favourite selection of books.
Algebraic Topology

M. Aguilar, S. Gitler, C. Prieto.
Algebraic Topology from a Homotopical Viewpoint,
Springer, 2002.

W.F. Basener.
Topology and Its Applications,
Wiley, 2006.

J.F. Davis, P. Kirk.
Lecture Notes in Algebraic Topology,
AMS, 2001.

A. Dold.
Lectures on Algebraic Topology,
Springer, 1980.

A. Hatcher.
Algebraic Topology,
Cambridge University Press, 2002.
Homepage of this book

W. Lück.
Algebraische Topologie: Homologie und
Mannigfaltigkeiten,
Vieweg, 2005.

W.S. Massey.
Algebraic Topology: an Introduction,
siebte Auflage, Springer, 1989.

W.S. Massey.
A Basic Course in Algebraic Topology,
dritte Auflage, Springer, 1997.
Hint: In this book, singular homology is based on cubes instead of simplices.

P. May.
A Concise Course in Algebraic Topology,
University of Chicago Press, 1999.

J. Strom.
Modern Classical Homotopy Theory,
American Mathematical Society, 2012.

T. tom Dieck.
Algebraic Topology,
European Mathematical Society,
2008.
Pointset Topology

K. Jänich.
Topologie,
achte Auflage, Springer, 2008.

J.L. Kelley.
General Topology, Springer, 1975.

J.R. Munkres.
Topology, zweite Auflage, Pearson, 2003.

L.A. Steen.
Counterexamples in Topology, Dover, 1995.
Homological Algebra

C. Weibel. An Introduction to Homological Algebra,
Cambridge University Press, 2008.
Category Theory

M. Brandenburg.
Einführung in die Kategorientheorie: Mit ausführlichen Erklärungen und zahlreichen Beispielen,
Springer Spektrum,
2015.

H. Herrlich, G.E. Strecker.
Category Theory,
dritte Auflage,
Sigma Series in Pure Mathematics,
Heldermann, 2007.

S. MacLane.
Categories for the Working Mathematician,
zweite Auflage, Springer, 1998.

B. Richter.
Kategorientheorie mit Anwendungen in Topologie,
Vorlesungsskript, WS 2010/11, Universität Hamburg,
pdf
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.
Exams
Please read the
information on organisation and formalities of this course.
Dates for the
oral exams (25 minutes):

February 12, 2016

March 2, 2016

March 17, 2016

April 6, 2016
Registration details:

Please register in FlexNow for the oral exam. Registration deadline: Two weeks before the exam.
(You can then deregister until one week before the exam).

Moreover, for the oral exam, you also need to register with Ms. Bonn
(M 217) for a time slot.

If applicable: For the Studienleistung (successful participation in the exercise classes), please register
in FlexNow before January 31.
.
If you prefer to take a combined oral exam on Algebraic Topology
I/II, please contact me.
Last change, February 5, 2016