Constraints on families of smooth 4-manifolds from Bauer-Furuta invariants

Abstract

We obtain constraints on the topology of families of smooth 4manifolds 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 Zpactions for any odd prime p, which can not 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.

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

“…In Section 3 we give constraints on smooth families of 4manifold using a finite-dimensional approximation of a families Pin − (2)-monopole map. Those constraints are analogues of some constraints by Baraglia [1] obtained from the families Seiberg-Witten monopole map. In Section 4 we give the proofs of Theorems 1.1 and 1.4: we shall construct concrete topological families of 4manifolds and show the non-smoothability of them using the constraints obtained in Section 3.…”

“…Then the family E is the fiberwise connected sum of E M and the trivialized bundle T n−1 × N → N . As in the proof of [1,Theorem 10.3], it is easy to see that w n−1 (H + (E M )) = 0. This non-vanishing together with (10) and (11) implies that…”

“…Proof. The proof is parallel to that of [1,Theorem 1.1]. Throughout this proof, the coefficients of cohomology groups are supposed to be…”

“…This argument extracts some homotopical difference between the homeomorphism groups and the diffeomorphism groups of some class of 4-manifolds. Soon after [7], using Seiberg-Witten theory for families in a different manner, D. Baraglia [1] extensively generalized the result in [7] on comparisons between the homeomorphism and diffeomorphism groups of 4-manifolds: he proved in [1, Corollary 1.9] that for every closed, oriented, simply-connected, smooth, and indefinite 4-manifold M with |σ(M )| > 8, the inclusion Diff(M ) ֒→ Homeo(M ) is not a weak homotopy equivalence. Here σ(M ) denotes the signature of M , and Diff(M ) and Homeo(M ) denote the groups of diffeomorphisms and homeomorphisms respectively.…”

“…The proof of this result by Baraglia is based on a finite-dimensional approximation of the families Seiberg-Witten monopole map. The purpose of this paper is to give analogues of arguments in [1] by Baraglia for the Pin − (2)-monopole equations introduced in [11], and to make use of the Pin − (2)-monopole analogues to generalize the above result by Baraglia on comparison between homeomorphism and diffeomorphism groups as follows:…”