Topological K-theory and vector fields on spheres

Topological K-theory and vector fields on spheres

Semester
WiSe 2024 / 25

Lecturer
Christoph Winges

Type of course (Veranstaltungsart)
Seminar

German title
Topologische K-Theorie und Vektorfelder auf Sphären

Contents
We will discuss topological K-theory, which is an invariant of topological spaces built in terms of
vector bundles over a given space. It turns out that the resulting (contravariant) functor is
homotopy invariant and satisfies an appropriate version of the excision theorem, and therefore gives
rise to a „generalised cohomology theory“. One of its key additional features is Bott
periodicity. After introducing topological K-theory, we will discuss some its classical
applications: one can determine the precise number of linearly independent vector fields over
spheres, and one can also show that there are no other real division algebras than the reals, the
complex numbers, the quaternions and the octonions.

Literature
Hatcher. Vector Bundles and K-theory; Knapp. Vektorbündel

Recommended previous knowledge
Analysis, Linear Algebra; some basic notions from topology (eg homotopy)

Time/Date
tba

Location
tba

Course homepage
https://homepages.uni-regensburg.de/~wic42659/
(Disclaimer: Dieser Link wurde automatisch erzeugt und ist evtl. extern)

Registration

Organisational meeting/distribution of topics: tba, more info will appear shortly on my
personal webpage. Feel free to send a mail to christoph dot winges at ur dot de in advance,
then I will inform you when more detailed information is available.
Registration for course work/examination/ECTS: FlexNow

Course work (Studienleistungen)

Presentation: Giving a seminar talk of roughly 90 minutes

Examination (Prüfungsleistungen)

Detailed written report of the seminar talk

Modules
BSem, MSem

ECTS
4,5 ECTS