Diffeomorphism Groups: algebra, topology, homology

If M is a smooth manifold, the set of all smooth self-maps of M with smooth inverses forms a group, denoted Diff(M), with a natural topology. This group -- in fact, even its identity component Diff_0(M) -- has an incredibly rich algebraic structure, rich enough to recover (at least in theory) M and its smooth structure. Understanding the algebraic structure of Diff(M) is an basic question in dynamics: a smooth action of a group G on M is just a homomorphism from G to Diff(M).

From a topological perspective, understanding the topology and (co)homology of Diff(M) gives us tools to classify fiber bundles with fiber M, flat connections on such M-bundles, certain foliations on manifolds, and spaces of metrics on M.

This summer workshop is designed to give graduate students interested in any aspect of this circle of ideas a taste of the study of diffeomorphism groups.

Minicourses

This course introduces classical and new results on the algebraic structure of diffeomorphism groups. These groups are algebraically simple (no nontrivial normal subgroups) -- for deep topological reasons due to Epstein, Mather, Thurston... but nevertheless have a very rich algebraic structure. We'll see that:

1. The algebraic structure of Diff(M) determines M
   If Diff(M) is isomorphic to Diff(N), then M and N are the same smooth manifold (Filipkiewicz)
2. The algebraic structure of Diff(M) "captures the topology" of Diff(M)
   Any group homomorphism from Diff(M) to Diff(N) is necessarily continuous. Any homomorphism from Homeo(M) to any separable group is necessarily continuos (Hurtado, Mann)

We'll explore consequences of these theorems and related results, as well as other fascinating algebraic properties of diffeomorphism groups of manifolds (for instance, distorted elements, left-invariant orders, circular orders...). In the last lecture, we'll touch on recent work on the geometry and metric structure of diffeomorphism groups.

Beginner -- assumend background knowledge.

1. Basic knowledge on smooth manifolds, Lie groups, and group actions, as in Lee's Introduction to smooth manifolds. The chapter on Lie groups and group actions is particularly recommended.

Intermediate level:

1. Ghys, Groups acting on the circle (survey paper)
2. Banyaga, The structure of classical diffeomorphism groups (book), chapter 1.

Advanced Level -- not seriously recommended until after the summer school!

1. Calegari, Circular groups, planar groups, and the Euler class .
2. Hurtado, Continuity of discrete homomorphisms of diffeomorphism groups

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

1. the homotopy type of Diff(M) when dim(M) < 4;
2. circle bundles, the Euler class, and the Milnor-Wood inequality; and
3. surface bundles, the Miller-Morita-Mumford classes, and Nielsen realization problems.

Beginner -- good background knowledge to have.

1. From Hatcher, Algebraic topology: basic knowledge of homotopy groups, fibrations, long exact sequence of homotopy groups
2. Hatcher, Vector bundles and K-theory Chapters 1 and 3.

Intermediate level:

1. Stasheff, Continuous cohomology of groups and classifying spaces (survey paper)
2. Hatcher, A 50 -Year View of Diffeomorphism Groups (talk notes)

Advanced Level

1. Morita, Geometry of characteristic classes (book)
2. Calegari, The Euler class of planar groups . This short research paper gives some nice connections between the two minicourses.