Homotopy theory of foliations and diffeomorphism groups

Overview: The PI’s plan is to combine the new homotopy theoretical methods that stem from the evolving field of the moduli space of manifolds with the classical foliation theory to study diffeomorphism groups. The synthesis of these techniques have already led the PI to find the analogue of Suslin’s rigidity theorem for surface diffeomorphism groups and to prove many non- vanishing results for characteristic classes of flat manifold bundles and to construct R~Z-invariants for such bundles in the spirit of secondary invariants for flat principal bundles. In 70s, Thurston used dynamical techniques to study the identity component of diffeomorphism groups in order to investigate the homotopy type of the Haefliger spaces. This research project is going in the opposite direction started from the PI’s theorem that the group homology of diffeomorphism groups and symplectomorphisms of surfaces in a range are isomorphic to the homology of certain infinite loop spaces. Therefore now we can translate what we know about the homotopy type of the Haefliger spaces to homological properties of diffeomorphism groups. In section 2, the PI outlined how one can translate many open problems regarding the (non)-vanishing of characteristic classes of flat bundles to a geometric problem about foliations that conceivably be studied via dynamical methods. In section 3, PI also mentions a surprising applica-tion of this method to revisit the generalized Smale’s conjecture. the PI shows that understanding the homotopy type of the identity component of the diffeomorphism groups for certain 3-manifolds is related to the homotopy type of a combinatorial complex associated to the three manifolds. In section 4 of the project description, the PI outlined few projects to revisit the homotopy type of the Haefliger space using the new homotopy theoretic methods. The first and the main problem in this part is to answer an old question of Haefliger that the h-principle theorem due to Bott and Segal for the continuous Lie algebra cohomology of vector fields should be a consequence of Mather-Thurston’s theorem for foliated manifold bundles. Intellectual merit: This ongoing research synthesizes techniques from homotopy theory, foliation theory, low dimensional topology and Lie algebra of infinite dimensional Lie groups to understand the homology of automorphism groups of manifolds as discrete groups. This point of view has led us to find surprising relations between open problems in the above-mentioned different fields. It is promising to study the Haefliger space via new methods in homotopy theory. In particular, the existence of the “smooth homotopy category” as it is recently popularized by Graeme Segal, should be the context that one could answer the above mentioned Haefliger’s question. Broader impact: The PI has shown his commitment to education and general public through variety of outreach activities, including teaching for three years in the national olympiad institute (YSC) in Iran, teaching AIME class at communication academy in Bay Area, mentoring in Graduate Research Opportunities for Women (GROW) at Northwestern. Regarding the research proposal, the PI believes people in the foliation community who study diffeomorphism groups as discrete groups are mainly in France and Japan. This grant would enhance the relation of the PI with their community in Europe and in Japan. The PI has given lecture series in Nantes on his results and will give series of talks in BΓ-school in Tokyo. The PI organized the joint Berkeley-Stanford seminar on diffeomorphism group, topology seminar at Northwestern