2020/10/01 by Hokuto Konno, Konno, Hokuto, M. Taniguchi +1
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.2010.00340
openalex publication_date 2020/10/01 · openalex created_date 2022/10/04 · openalex updated_date 2026/07/28
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,∂) \hookrightarrow 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) \hookrightarrow 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.