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

Abstract

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

Diffeomorphisms of 4-manifolds with boundary and exotic embeddings

“…To our knowledge, there is no known counter example to this conjecture. Positive evidence to the non-surjectivity of the maps is given in [36] where nonsurjectivity is shown that for certain W , which is a partial generalization of results for closed 4-manifolds in [9,15,21,22,35,46] to 4-manifolds with boundary.…”

The groups of diffeomorphisms and homeomorphisms of 4-manifolds with boundary

“…There are few known examples of closed smooth 4-manifolds X for which (60) are not surjections: the first example is a K3 surface by Donaldson [13], and in fact so is every homotopy K3 surface, which one can check using a result by Morgan and Szabó [37]. It follows from Baraglia's constraint [2, Theorem 1.1] that an Enriques surface is also an example, and so is a stabilization of an Enriques surface by the connected sum with some nonsimply-connected 4-manifolds by Nakamura and the first author [24,Corollary 1.6] Example 4.7. We consider the Brieskorn homology 3-sphere Σ(p, q, r) for a pairwise relatively prime triple of positive integer (p, q, r).…”

“…This is just because there is no obvious way to define a local system on Th(W ) induced from a local system on the base space B. To use relative cohomologies, we need to have a map between pairs f : (D(W 0 ), S(W 0 )) → (W 1 , W 1 \ {0}) × (N, L) instead of (24). But we could not figure out whether we can obtain such a map as the families relative Bauer-Furuta invariant, because it seems essential to cut the domain of f by the compact sets K1 , K2 in Lemma 2.13 to obtain an appropriate index pair (N, L).…”