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Seminar on the h-cobordism theorem SemesterWiSe 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.
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 |