Diffeomorphism groups in dimensions 2 and 3

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The proposal is devoted to diffeomorphisms groups of manifolds of thedimension 2 and 3. In the dimension 2 the main problem is to understand thealgebraic structure of the groups of isotopy classes of suchdiffeomorphisms, known as mapping class groups of surfaces or Teichmuellermodular groups. The proposal addresses some of the central problems aboutthe mapping class groups: understanding the structure of the Torelli groups,braid groups of surfaces, and mapping class groups themselves. The problemsaddressed in the proposal arethe ones which either hold the key for the fundamental propertiesof the groups in question (like the problem of finite presentability of theTorelli groups) or have connections with other branches of mathematics (likethe problem of understanding the Wajnryb presentations from the point ofview of algebraic K-theory and topological quantum field theory). In thedimension 3 the key problem is the understanding of the homotopy type of thediffeomorphisms groups, especially of their connected components. The maingoal here is to relate the few available results in this area to the moremainstream tools of topology and to extend, on this basis, these results toother manifolds. A particular goal is to understand the diffeomorphismsgroups for all lens spaces, the simplest 3-manifolds after spheres.The proposal belongs to the field of low dimensional topology, whichinvestigates the possible shapes of objects (namely, the so-calledmanifolds) in dimension 2 and 3. These shapes continue to be in the focus ofthe mathematical research for more than a hundred years. In the lastdecades, the theory of 2-dimensional shapes, i.e. surfaces, acquired anadditional significance as a result of a promising physical "theory ofeverything ", namely the so-called string theory. In this theory, theelementary constituents of matter are not point-like particles, but rathercollections of loops ("strings") sweeping a surface during their motion. The3-dimensional shapes provide the model for the large-scale structure of theuniverse. Some previous results of the PI in this direction have alreadyattracted the attention of physicists working in general relativity. In bothdimensions, 2 and 3, one of the key problem is to understand the possiblesymmetries of these shapes. The proposal is devoted to some of the mainaspects of this problem.