Simple-homotopy theory and the s-cobordism theorem

Simple-homotopy theory and the s-cobordism theorem

Semester
WiSe 2022 / 23

Lecturer
Christoph Winges

Type of course (Veranstaltungsart)
Seminar

German title
Einfache Homotopietheorie und der s-Kobordismussatz

Contents
Simple-homotopy theory asks whether any homotopy equivalence of finite CW-complexes can be expressed
as a composition of some elementary geometric moves on the cells (called elementary expansions and
elementary collapses). It turns out that the answer to this question is highly dependent on the
fundamental group, and the non-simpleness of a homotopy equivalence can be detected by an element of
the Whitehead group of the fundamental group, which is a certain K-theoretic invariant. This
provides for example tools to distinguish lens spaces which are homotopy equivalent, but not
homeomorphic. In the later talks of the seminar, we will see how this theory comes to bear in
understanding h-cobordisms between closed manifolds, ie cobordisms with the property that the
inclusion of each boundary component is a homotopy equivalence. The s-cobordism theorem asserts that
such an h-cobordism is isomorphic to a cylinder on one of the boundaries precisely if the inclusion
of that boundary component is a simple homotopy equivalence.

Literature
Please consult the seminar programme, which will be available shortly on the GRIPS page of the
seminar.

Recommended previous knowledge
Algebraic Topology, for the later talks also some Differential Topology. Depending on the
attendees’ knowledge of these topics, some of the necessary prerequisites can also be supplied
in the form of overview talks.

Time/Date
Tue 16-18

Location
M 103

Course homepage
https://elearning.uni-regensburg.de/course/view.php?id=57586
(Disclaimer: Dieser Link wurde automatisch erzeugt und ist evtl. extern)

Registration

Organisational meeting/distribution of topics: 27th July 2022, 2.15 pm; location tba on GRIPS
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, MV, MSem

ECTS
BSem und MSem: 4,5 LP bei Studienbeginn ab WS 15/16, 6 LP bei Studienbeginn vor WS 15/16.
LA-GySem: 6 LP. MV und Nebenfach: 4,5 LP bei Studienbeginn ab WS 15/16, 6 LP bei Studienbeginn vor
WS 15/16 +++ weitere Details: siehe Modulkatalog +++