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.