The groups of diffeomorphisms and homeomorphisms of 4-manifolds with boundary
Hokuto Konno, Masaki Taniguchi

TL;DR
This paper uses advanced invariants from Seiberg-Witten theory to show that the groups of diffeomorphisms and homeomorphisms of certain 4-manifolds with boundary are not homotopy equivalent, revealing new topological constraints.
Contribution
It introduces new constraints on smooth families of 4-manifolds with boundary using Manolescu's invariants, extending previous results to manifolds with boundary.
Findings
The inclusion map from diffeomorphisms to homeomorphisms is not a weak homotopy equivalence under certain conditions.
Constraints generalize previous results on closed 4-manifolds and boundary cases.
Provides obstructions to smooth structures on 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 with boundary with an assumption on the Fr{\o}yshov invariant or the Manolescu invariants of , the inclusion map 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 is also not a weak homotopy equivalence under the same assumption on $\partial…
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Taxonomy
TopicsGeometric and Algebraic Topology · Homotopy and Cohomology in Algebraic Topology · Mathematical Dynamics and Fractals
