Homotopy Theory of Foliations and Diffeomorphism Groups

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Foliation theory is a field of mathematics, which is roughly 50 years old, whose object of study is certain decomposition of manifolds into path-connected subsets, called leaves. A foliation looks locally like a decomposition of the manifold as a union of parallel submanifolds of lower dimensions. Such geometric structures naturally arise in physics and geology. And in mathematics the depth and breadth of foliated objects made mathematicians use tools from many different branches of mathematics including differential geometry, homotopy theory, noncommutative geometry, ergodic theory and dynamical systems. The PI intends to use new tools from homotopy theory to investigate the relation between foliations and diffeomorphism groups.The existence and classification of foliations and the implication of such structures on the global topology of manifolds have been extensively studied in the past five decades. However, there are still many mysteries, perhaps the most important of which in the homotopy theory of foliation is the Haefliger conjecture. Haefliger asked whether all plane fields on a manifold whose dimensions are roughly less than the half of the dimension of the manifold are integrable up to homotopy. It was shown by Mather and Thurston that the homotopy theory of foliations is naturally related to the homological invariants of the diffeomorphism groups made discrete. But the group homologies of diffeomorphism groups as discrete groups tend to be very large and are poorly understood. On the other hand diffeomorphism group with the Whitney topology is better understood, in particular, Galatius and Randal-Williams' program developed new tools to study the classifying space of these groups with the Whitney topology. 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 homological invariants of diffeomorphism groups made discrete.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.