The groups of diffeomorphisms and homeomorphisms of 4-manifolds with boundary

Abstract

We give constraints on smooth families of 4-manifolds with boundary using Manolescu's Seiberg-Witten Floer stable homotopy type, provided that the fiberwise restrictions of the families to the boundaries are trivial families of 3-manifolds. As an application, we show that, for a simply-connected oriented compact smooth 4-manifold X with boundary with an assumption on the Frøyshov invariant or the Manolescu invariants α, β, γ of ∂X, the inclusion map Diff(X, ∂) ֒→ Homeo(X, ∂) between the groups of diffeomorphisms and homeomorphisms which fix the boundary pointwise is not a weak homotopy equivalence. This combined with a classical result in dimension 3 implies that the inclusion map Diff(X) ֒→ Homeo(X) is also not a weak homotopy equivalence under the same assumption on ∂X. Our constraints generalize both of constraints on smooth families of closed 4-manifolds proven by Baraglia and a Donaldson-type theorem for smooth 4-manifolds with boundary originally due to Frøyshov.

Diffeomorphisms of 4-manifolds with boundary and exotic embeddings

“…To our knowledge, there is no known counter example to this conjecture. Positive evidence to the non-surjectivity of the maps is given in [36] where nonsurjectivity is shown that for certain W , which is a partial generalization of results for closed 4-manifolds in [9,15,21,22,35,46] to 4-manifolds with boundary.…”

Seiberg-Witten Floer spectra for $b_1>0$

“…Furthermore, we expect that it should be possible to consider more detailed applications to the question of when a family of three-manifolds extends to a family of 4-manifolds with boundary. Compare with recent work of Konno-Taniguchi [KT20] in the case that the boundary family of 3-manifolds is the trivial family of a rational homology sphere.…”

Involutions, knots, and Floer K-theory

“…Recall the comuputation of Manolescu's α invariant of Y (see [55, Proposition 3.8, Page 172]):α(Σ(2, 3, 12k + 1)) = 0, α(−Σ(2, 3, 12k + 1)) = −γ(α(Σ(2, 3, 12k + 1))) = 0, α(−Σ(2, 3, 12k − 1)) = −γ(α(Σ(2, 3, 12k − 1))) = 0.Thus in any case we have α(Y ) = 0. Then it follows from[44, Theorem 1.2] that −σ(W )/8 ≤ α(Y ) = 0, but since σ(W ) = −16, this is a contradiction.Bryan proved in [12, Theorem 1.8] that, for a spin odd involution ι on a spin rational cohomology K3 surface X, we have b + ι (X) = 1. If f in Proposition 5.4 is an involution ι, then Proposition 5.4 implies that b + ι (W ) = 1 or b + ι (W ) = 3.…”