Geometry and Topology Seminar: "The rational strong Novikov conjecture, groups of diffeomorphisms, and symmetric Hilbert-Hadamard spaces"

Abstract: The rational strong Novikov conjecture is a deep problem in noncommutative geometry. It implies important conjectures in manifold topology and differential geometry such as the (classical) Novikov conjecture on higher signatures and the Gromov-Lawson conjecture on positive scalar curvature. Using C*-algebraic and K-theoretic tools, we prove that the rational strong Novikov conjecture holds for any discrete group admitting an isometric and proper action on an admissible Hilbert-Hadamard space, which is a (typically infinite-dimensional) generalization of complete simply connected nonpositively curved Riemannian manifolds. In particular, a prominent example of an admissible Hilbert-Hadamard space is the space of L^2-Riemannian metrics on a smooth manifold with a fixed density. This space can be viewed as an infinite-dimensional symmetric space. As a result, our result implies the rational strong Novikov conjecture holds for geometrically discrete subgroups of the group of volume preserving diffeomorphisms of a closed smooth manifold. This is joint work with Sherry Gong and Guoliang Yu.

Host: Michael Landry