Bauer-Furuta invariants under \({\mathbb{Z}_2}\)-actions (Q1014234)

Seiberg-Witten invariants are integers associated to a smooth 4-manifold together with a \(\text{Spin}^c\) structure, which may be interpreted as a signed count of zeroes of a certain Fredholm map (called the monopole map). Using finite-dimensional approximation of the monopole map, \textit{S. Bauer} and \textit{M. Furuta} defined a stable cohomotopy refinement of the Seiberg-Witten invariants [Invent. Math. 155, No.~1, 1--19 (2004; Zbl 1050.57024)]. The main theorem in the paper under review states that the Bauer-Furuta invariants vanish for 4-manifolds admitting a smooth \({\mathbb Z}_2\)-action, under certain additional conditions. Explicit locally linear \({\mathbb Z}_2\)-actions on the connected sum of two copies of the \(K3\) surface, and on \(K3\) itself, are constructed using a method of Edmonds and Ewing and shown to be nonsmoothable (i.e. not conjugate by a homeomorphism to a smooth action).