#### Volume 21, issue 1 (2021)

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Constraints on families of smooth $4$–manifolds from Bauer–Furuta invariants

### David Baraglia

Algebraic & Geometric Topology 21 (2021) 317–349
##### Abstract

We obtain constraints on the topology of families of smooth $4$–manifolds arising from a finite-dimensional approximation of the families Seiberg–Witten monopole map. Amongst other results these constraints include a families generalisation of Donaldson’s diagonalisation theorem and Furuta’s $\frac{10}{8}$ theorem. As an application we construct examples of continuous ${ℤ}_{p}$–actions, for any odd prime $p$, which cannot be realised smoothly. As a second application we show that the inclusion of the group of diffeomorphisms into the group of homeomorphisms is not a weak homotopy equivalence for any compact, smooth, simply connected, indefinite $4$–manifold with signature of absolute value greater than $8$.

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##### Keywords
$4$–manifolds, Seiberg–Witten, Bauer–Furuta
##### Mathematical Subject Classification 2010
Primary: 57R57
Secondary: 57R22, 57R50