The friday fish seminar on the Madsen-Weiss theorem

The diffeomorphism group Diff(M) of a manifold M is a scary beast: It is an infinite dimensional topological group. Automorphisms of mathematical objects are inherently interesting, but this group also shows up in practical application. The diffeomorphism group of surfaces is for example closely related to the classification of three dimensional manifolds via Heegaard splittings, and the classifying space BDiff(M) of the diffeomorphism group classifies smooth M-bundles over reasonable topological spaces. It is in general difficult to determine when a bundle is trivial. Obstructions to triviality are given by pulling back cohomology classes of BDiff(M) along the classifying map. It is therefore desirable to compute the cohomology of BDiff. The Madsen-Weiss Theorem is a computation of the cohomology of BDiff(M) in the stable range. The goal of the seminar is to understand the proof of the Madsen-Weiss theorem.

The seminars take place in room 610 of the Hans Freudenthal gebouw in Utrecht. The seminars start at 15:00 and last till 17:00 or so.

Schedule

Date Name Topics
19-10-2018 Thomas Rot Introductory talk. Diffeomorphism groups, classifying spaces, characteristic classes. The scanning map (Section 1 [1])
02-11-2018 Alvaro Del Pino Gomez Gremain's proof of the Earle-Eells Theorem: Appendix B of [1], see also Chapters 7-10 in [3]. If time permits some discussion on Dehn twists and the Mapping class group.
09-11-2018 Lennart Meier The group completion theorem: Appendix D of [1]. Given a topological group G, it is well-known that the loop space of the classifying space of G is equivalent to G again. The same is true for topological monoids if we demand that the monoid of components is actually a group. For a general topological monoid M, the loops on the classifying space will produce a “group completion” of M, which means that the homology changes in a predictable way, while the homotopy groups might change more drastically. The precise statement is the content of the group completion theorem, of which we will present a proof. The main technical tool will be the theory of quasi-fibrations.
23-11-2018 Luca Accornero Madsen-Weiss I: Section 2 and part of Section 3 of [1]
11-12-2018 (11:00!) Aldo Witte Madsen-Weiss II: Deloopings. Remainder section 3
11-01-2019 Alvaro del Pino Gomez Madsen-Weiss III: More deloopings (the hard case)
25-01-2018 Jack Davies Computation of the stable homology. Appendix C of [1]

Literature

  1. Hatcher: A short exposition of the Madsen Weiss Theorem
  2. Eliashberg, Galatius, Mishachev: Madsen-Weiss for geometrically minded topologists
  3. Kupers: Lectures on diffeomorphism groups of manifolds
  4. Randal-Williams Homological stability for unordered configuration spaces
  5. Wahl: Homological stability for mapping class groups of surfaces