Families of diffeomorphisms, embeddings, and positive scalar curvature metrics via Seiberg-Witten theory

Abstract

We construct infinite rank summands isomorphic to ${\mathbb{Z}}^{\infty }$ in the higher homotopy and homology groups of the diffeomorphism groups of certain $4$-manifolds. These spherical families become trivial in the homotopy and homology groups of the homeomorphism group; an infinite rank subgroup becomes trivial after a single stabilization by connected sum with $S^2\times S^2$. The stabilization result gives rise to an inductive construction, starting from non-isotopic but pseudoisotopic diffeomorphisms constructed by the second author in 1998. The spherical families give ${\mathbb{Z}}^{\infty }$ summands in the homology of the classifying spaces of specific subgroups of those diffeomorphism groups. The non-triviality is shown by computations with family Seiberg-Witten invariants, including a gluing theorem adapted to our inductive construction. As applications, we we obtain infinite generation for higher homotopy and homology groups of spaces of embeddings of surfaces and $3$-manifolds in various $4$-manifolds, and for the space of positive scalar curvature metrics on standard PSC $4$-manifolds.

60 pages, 1 figure