2025/01/21 by Dave Auckly, Auckly, Dave, Daniel Ruberman +1 · 1 citation
Physics and Astronomy · Mathematics · #Advanced Differential Geometry Research #Black Holes and Theoretical Physics #Geometric Analysis and Curvature Flows
paper · pdf · doi:10.48550/arxiv.2501.11892
We construct infinite rank summands isomorphic to ℤ^∞ 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 S2 × S2. 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 ℤ^∞ 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.