SPACES OF DIFFEOMORPHISMS OF DISKS AND CHARACTERIS

Spaces of diffeomorphisms of disks and characteristic classes

The proposed research is about symmetries of the n-disk, the region D(n) in n-dimensional euclidean space consisting of all points having distance at most 1 from a specified origin. The symmetries under consideration are invertible differentiable transformations from D(n) to D(n) which take every point on the boundary of D(n) to itself. The goal is to understand some of the geometric and qualitative properties of the sets Diff(D(n)) made up by all such symmetries. Itis a fact that Diff(D(n)) is a crucial building block in all kinds of sets of differentiable symmetries.For the sort of measurements that this research aims for, there is a reduction, available since around 1970, which predicts the outcomes of such measurements made on Diff(D(n)) in terms of outcomes of similar measurements made on another set of symmetries, Top(n). The set Top(n) consists of invertible continuous (not necessarily differentiable) transformations from euclidean n-space to itself. An advantage of this reduction is that the variability of the input n in Top(n) can be used to good effect. It is easy to relate Top(n) to Top(n+1), because a transformation from euclidean n-space to itself determines a transformation from euclidean (n+1)-space to itself which leaves the last coordinate of any point in (n+1)-space unchanged. It is also possible to make sense of Top(n) when n is infinity. Moreover, the geometry and qualitative properties of Top(infty) are in some respects quite well understood, much better in many ways than the geometry and qualitative properties of Top(n) for finite n. The exploration of Top(infty) was one of the major mathematical enterprises in the second half of the 20th century. In addition to many beautiful geometric ideas it also led to a number of new algebraic concepts. Part of the strategy proposed here for understanding some of the geometry of Top(n), and therefore some of the geometry of Diff(D(n)), is a calculus approach.The idea is to think of Top(n) as a function of n and to play calculus with that function, to ask questions about rates of change. For example, the extent to which the continuous map from Top(n) to Top(n+1) just described deviates from being a homotopy equivalence would be filed under first rate of change aspects. (Algebraic topology has well-established concepts of equivalence and efficient ways to quantify how much a continuous map deviates from being an equivalence.) More to the point, one hopes to understand Top(n), for a fixed n which may not be very large, in terms of Top(infty) and the rates of change of Top(k) for very large k. This is analogous to how the value of an analytic function such as cos(x), for any real number x, may be expressed in terms of the value, first derivative and higher derivatives of the function at x=0. A solid framework for doing that kind of calculus with functions whose values are geometric objects rather than numbers has been developed in the last two decades. It is now called functor calculus and the functions in question are normally called functors . The principal investigator is not the originator of that framework, but has made substantial contributions to it. In our example, where the functor takes a positive integer n to Top(n), the value Top(infty) of the functor at infinity is well understood in many respects, as noted earlier. The first derivative of the functor at infinity is also rather well understood; this is a more recent development in which, again, the principal investigator has some part. The second derivative of the function at infinity is not yet well understood but it seems ripe for a thorough investigation. Such an investigation, if successful, will almost automatically give significant new results on Top(n) and Diff(D(n)).