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

DOI10.1016/J.AIM.2022.108627zbMATH Open1512.57045arXiv2010.00340WikidataQ113880842 ScholiaQ113880842MaRDI QIDQ6350308

Hokuto Konno, Masaki Taniguchi

Publication date: 1 October 2020

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 Fr{o}yshov invariant or the Manolescu invariants $$ of $partialX$, the inclusion map $mathrmDiff(X,partial)hookrightarrowmathrmHomeo(X,partial)$ 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 $mathrmDiff(X)hookrightarrowmathrmHomeo(X)$ is also not a weak homotopy equivalence under the same assumption on $partialX$. 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 Fr{o}yshov.

Mathematics Subject Classification ID

Stable homotopy theory, spectra (55P42) Applications of global analysis to structures on manifolds (57R57) Differential topological aspects of diffeomorphisms (57R50) General topology of 4-manifolds (57K40)

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