Diffeomorphism groups in dimensions 2 and 3

Michigan State University, East Lansing MI

Investigators

Abstract

The proposal is devoted to diffeomorphisms groups of manifolds of the dimension 2 and 3. In the dimension 2 the main problem is to understand the algebraic structure of the groups of isotopy classes of such diffeomorphisms, known as mapping class groups of surfaces or Teichmueller modular groups. The proposal addresses some of the central problems about the mapping class groups: understanding the structure of the Torelli groups, braid groups of surfaces, and mapping class groups themselves. The problems addressed in the proposal are the ones which either hold the key for the fundamental properties of the groups in question (like the problem of finite presentability of the Torelli groups) or have connections with other branches of mathematics (like the problem of understanding the Wajnryb presentations from the point of view of algebraic K-theory and topological quantum field theory). In the dimension 3 the key problem is the understanding of the homotopy type of the diffeomorphisms groups, especially of their connected components. The main goal here is to relate the few available results in this area to the more mainstream tools of topology and to extend, on this basis, these results to other manifolds. A particular goal is to understand the diffeomorphisms groups for all lens spaces, the simplest 3-manifolds after spheres. The proposal belongs to the field of low dimensional topology, which investigates the possible shapes of objects (namely, the so-called manifolds) in dimension 2 and 3. These shapes continue to be in the focus of the mathematical research for more than a hundred years. In the last decades, the theory of 2-dimensional shapes, i.e. surfaces, acquired an additional significance as a result of a promising physical "theory of everything ", namely the so-called string theory. In this theory, the elementary constituents of matter are not point-like particles, but rather collections of loops ("strings") sweeping a surface during their motion. The 3-dimensional shapes provide the model for the large-scale structure of the universe. Some previous results of the PI in this direction have already attracted the attention of physicists working in general relativity. In both dimensions, 2 and 3, one of the key problem is to understand the possible symmetries of these shapes. The proposal is devoted to some of the main aspects of this problem.