This project is a multi-year research plan to develop new tools for the study of the structure of transformation groups and mapping class groups. The transformation groups such as diffeomorphism groups and homeomorphism groups of a manifold describe all symmetries of the manifold. A manifold is a space that is locally a Euclidean space. The surface of the earth, the surface of a donut and our three or four dimensional universe are all examples of manifolds. The mapping class group is a reduced form of the homeomorphism group of a manifold, in which two transformations are identified if they can be connected by a path of transformations. These objects have connections with many areas of mathematics, including geometric topology, geometric group theory, dynamics, number theory, quantum field theory, representation theory, and algebraic geometry. This project studies a number of questions about the structure of symmetries of manifolds. Broader impacts of these efforts include reading groups for graduate students.
Previously, in a joint work with Kathryn Mann, the principal investigator has proved Ghys' dimension conjecture that given two manifolds M and N, if the homeomorphism group of M has a nontrivial homomorphism to the homeomorphism group of N, then the dimension of M must be less than or equal to the dimension of N. To prove this, an "Orbit Classification Theorem" is developed, which says that every orbit of such a homomorphism is homeomorphic to a cover of some configuration space of M. This finding has a potential to give more results about classifying homomorphisms between transformation groups. Since we have figured out all orbits, now the challenge is to understand how to glue those orbits inside the manifold N. Another line of work studies the projection map from the transformation group to the mapping class group. In particular, we ask which subgroups of the mapping class group have sections under the projection, where the existence of sections implies that the surface bundle that the subgroup determines will admit a flat structure. In previous work on this subgroup question the principal investigator has proved several results using techniques of "hidden torsion" and rotation number. More tools will be developed specifically to attack this question. Another direction to explore concerns diffeomorphism groups of high dimensional manifolds, specifically, whether there is an example of an infinite torsion group that acts faithfully on some manifold.
This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.
