Cours spécialisé (GT,TA)
Danica Kosanovic
Contact : danica.kosanovic à imj-prg.fr
Pas de notes de cours prévues.
Langue du cours : English
Présentation
This will be an introduction to geometric topology, a field of mathematics concerned with topological properties of manifolds. We will prove some fundamental results about smooth manifolds of dimension 5 or higher -- like the Schoenflies theorem, the generalised Poincaré conjecture, the existence of exotic smooth structures -- several of which have been awarded with Fields medals. In the second part of the course we will see how much of this extends to dimension 4.
Contenu
- Recollections from differential topology (integration of vector fields, transversality, submanifolds, regular homotopy)
- Handles, geometric and algebraic intersection numbers. Handle calculus: reordering, cancelling, turning upside-down.
- Whitney trick, Morse chain complex. Handle slides, the proof of the s-cobordism Theorem.
- Whitehead torsion, corollaries.
- Freedman's s-cobordism Theorem in dimension 4.
- Exotic smooth structures.
Prérequis
All GT and TA courses can be useful, but especially important are Homotopie I and II, Topologie algébrique I and II. I recommend having a look at the first few chapters of [Wall].
Bibliographie
- Wall, C. T. C. Differential Topology. Cambridge Studies in Advanced Mathematics 156. Cambridge: Cambridge University Press (2016)
- Kosinski, A. Differential manifolds. Pure and Applied Mathematics, 138. Academic Press, Inc., Boston, MA (1993)
- Freedman, M. H.; Quinn, F. S. Topology of 4-manifolds. Princeton Mathematical Series, 39. Princeton, NJ: Princeton University Press (1990)
- Behrens, S.; Kalmár, B.; Kim, M. H.; Powell, M.; Ray, A. The disc embedding theorem. Oxford: Oxford University Press (2021)