Minicourse: Topological aspects of diﬀeomorphism groups. Abstract. The cohomology of the diﬀeomorphism group Diﬀ(M) of a manifold M and its classifying space BDiﬀ(M) are important to the study of ﬁber bundles with ﬁber M. In particular, we can learn a lot about M bundles by (1) ﬁnding nonzero elements of H∗(BDiﬀ(M)) and (2) relating these classes to the topology/geometry of individual bundles. A good start to (1) is to understand the topology of Diﬀ(M), and this has been done in low dimensions (by Smale, Hatcher, Earle-Eells, Gabai, and others). An example of (2) is the study of ﬁber bundles admitting a ﬂat connection (as pioneered by Milnor and Morita). This course will discuss (1) and (2) through a few rich examples and in connection to major areas of current research. Our discussion will include (a) the homotopy type of Diﬀ(M) when dim(M) < 4; (b) circle bundles, the Euler class, and the Milnor-Wood inequality; and (c) surface bundles, the Miller- Morita-Mumford classes, and Nielsen realization problems.
Disclaimer: These are notes are from lectures given at a graduate student workshop on diﬀeomorphism groups at UC Berkeley in June 2015. They are not meant as a source for details or as a primary reference. Unless otherwise noted, all objects considered below are oriented (manifolds, diﬀeomorphisms, bundles, etc). Usually this is omitted from the notation. For example, Diﬀ(M) always means orientation-preserving + diﬀeomorphisms, and I write GLn(R) when sometimes I mean GLn (R).
1 Lecture 1. The Euler class
I. Introduction. Fiber bundles M → E → B where M,B smooth manifolds.
S Locally trivial, globally interesting, E = Uα × M/ ∼
φαβ : Uα ∩ Uβ → Homeo(M). Scanned by CamScanner Examples Covering spaces, vector bundles, circle bundles, surface bundles
Problem Distinguish bundles.
Characteristic classes (measure global nontriviality)
• M manifold
• G ⊂ Homeo(M), e.g. Diﬀ(M), Symp(M, ω), Cont(M, α), Lie group
Deﬁnition A characteristic class c is an assignment