Constraints on families of smooth 4-manifolds from $\mathrm{Pin}^{-}(2)$-monopole

TL;DR

This paper extends previous results on the topology of 4-manifold diffeomorphism groups using $ ext{Pin}^{-}(2)$-monopole equations, providing new examples where smooth and topological automorphisms differ.

Contribution

It generalizes Baraglia's results by employing $ ext{Pin}^{-}(2)$-monopole equations and introduces new examples of 4-manifolds with non-surjective $ ext{Diff}$ to $ ext{Homeo}$ maps.

Findings

Inclusion $ ext{Diff}(X) o ext{Homeo}(X)$ is not a weak homotopy equivalence for many 4-manifolds.

New examples of 4-manifolds where $ ext{Diff}(X)$ and $ ext{Homeo}(X)$ differ on $ ext{pi}_0$.

Abstract

Using the Seiberg-Witten monopole equations, Baraglia recently proved that for most of simply-connected closed smooth $4$-manifolds $X$, the inclusions $\mathrm{Diff}(X) \hookrightarrow \mathrm{Homeo}(X)$ are not weak homotopy equivalences. In this paper, we generalize Baraglia's result using the $\mathrm{Pin}^{-}(2)$-monopole equations instead. We also give new examples of $4$-manifolds $X$ for which $\pi_{0}(\mathrm{Diff}(X)) \to \pi_{0}(\mathrm{Homeo}(X))$ are not surjections.