Universität Regensburg   IMPRESSUM   DATENSCHUTZ
Fakultät für Mathematik Universität Regensburg
Algebraic Topology II
Semester
SoSe 2022

Lecturer
Pavel Sechin

Type of course (Veranstaltungsart)
Vorlesung

German title
Algebraische Topologie II

Contents
The focus of this course will be on the homotopy category of topological spaces and various tools that allow to study it. In particular, we will introduce higher homotopy groups (where higher refers to the fundamental group being the first homotopy group) and 2-categorical tools that allow to apply (2-)categorical constructions for the study of the homotopy world. The abstract nonsense part of the course will be complemented by the geometric introduction into vector bundles and Serre spectral sequence. The latter is an effective tool for the computation of various (co)homology groups.

The following topics will be covered (* if time permits):
  • higher homotopy groups, fibrations and cofibrations;
  • Hurewicz theorem, Whitehead theorem, Freudenthal theorem, *Blackers-Massey theorem;
  • vector bundles, *principal G-bundles and their classifying spaces, characteristic classes;
  • the Serre spectral sequence.


Literature

T. tom Dieck, Algebraic Topology, Vol. 8. European Mathematical Society, 2008.

A. Hatcher, Algebraic Topology, 2001.

M. Mather, Pull-backs in Homotopy Theory, Canadian Journal of Mathematics, 28(2), 225-263 (1976).

J. Milnor and J.D. Stasheff, Characteristic Classes, Annals of mathematics studies 76, 1975.

A. Hatcher, Vector bundles and K-theory, 2003.

and other.

Recommended previous knowledge
Algebraic Topology I, basics of category theory (e.g. Yoneda lemma, limits and colimits, equivalences of categories)

Time/Date
Wed 12-14, 16-18 (exercises); Fr 14-16

Location
M 101 / M 009 (exercises)

Registration
  • Registration for the exercise classes: via GRIPS
  • Registration for course work/examination/ECTS: FlexNow
Course work (Studienleistungen)
  • Successful participation in the exercise classes: 50% of points in the exercises, presentation
    of a solution in class
Examination (Prüfungsleistungen)
  • Oral exam: Duration: 30 minutes, Date: first week after the lecture period, re-exam: Date: by
    individual appointment
Modules
BV, MV, MArGeo, MGAGeo

ECTS
9