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Fakultät für Mathematik Universität Regensburg
Seminar on the h-cobordism theorem
Semester
WiSe 2019 / 20

Lecturer
Bernd Ammann, Raphael Zentner

Type of course (Veranstaltungsart)
Seminar

Contents
A bordism is a (differentiable) compact manifold W with boundary M, such that M is the disjoint union of two manifolds M1 and M2. We suppose that the inclusions M1 → W and M2 → W are homotopy equivalences. The h-cobordism theorem states that then W is diffeomorphic to a cylinder M1 x [0,1]. The main goal of the seminar is to prove this theorem, based on Morse theoretic methods.

We then show important applications, in particular the Poincaré conjecture in higher dimensions: Each simply-connected compact differentiable manifold without boundary of dimension n ≥ 5 with the integral homology of a sphere is already homeomorphic to a sphere. Such spaces are even diffeomorphic to a sphere in the cases n=5 and n=6 (not proven in the seminar), but e.g. in dimension n=7 there are manifolds which are homeomorphic, but not diffeomorphic to a sphere. Such manifolds are called exotic spheres.

Literature
John Milnor: Lectures on the h-cobordism theorem
Michèle Audin, Mihai Damian: Morse theory and Floer Homology

Recommended previous knowledge
Foundations of differential geometry. Some knowledge about Morse theory is helpful, but not necessary

Time/Date
Monday 16-18

Location
M103

Course homepage
http://www.mathematik.uni-regensburg.de/ammann/lehre/2019w_h-cobordismus/
(Disclaimer: Dieser Link wurde automatisch erzeugt und ist evtl. extern)

Registration

  • Organisational meeting/distribution of topics: Monday, July 22, 16.15 (M 201)
  • 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
Regelungen bei Studienbeginn vor WS 2015 / 16
  • Benotet:
    • O. g. Studienleistung und o. g. Prüfungsleistung; die Note ergibt sich aus dem Seminarvortrag
  • Unbenotet:
    • O. g. Studienleistung
Modules
BSem, MV, MSem, LA-GySem

ECTS
Siehe Modulkatalog. MV und Nebenfach: 4,5 LP bei Studienbeginn ab WS 15/16, 6 LP bei Studienbeginn
vor WS 15/16
Druckansicht