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

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 10 8 theorem. As an application we construct examples of continuous 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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$4$–manifolds, Seiberg–Witten, Bauer–Furuta
Mathematical Subject Classification 2010
Primary: 57R57
Secondary: 57R22, 57R50
Received: 22 July 2019
Revised: 11 February 2020
Accepted: 8 May 2020
Published: 25 February 2021
David Baraglia
School of Mathematical Sciences
The University of Adelaide
Adelaide SA