The Dehn twist on a sum of two K3 surfaces

Abstract

Ruberman gave the first examples of self-diffeomorphisms of four-manifolds that are isotopic to the identity in the topological category but not smoothly so. We give another example of this phenomenon, using the Dehn twist along a 3-sphere in the connected sum of two K3 surfaces.

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

“…One may check the same statement holds also for any homotopy K3 surface using the Seiberg-Witten invariants and a result by Morgan and Szabó [9]. We note that examples of 4manifolds X for which π 0 (Diff(X)) → π 0 (Homeo(X)) are not injections are known a little more: the first example was given by Ruberman [13], and later additional examples were given by Baraglia and the first author [2], and by Kronheimer and Mrowka [8] recently.…”

“…Let X be as in the statement of Theorem 1.4 and M ′ = mS#M . Define N by (8). Recall that an Enriques surface S is homeomorphic to −E 8 #(S 2 × S 2 )#W , where W is a non-spin topological rational homology 4-sphere with π 1 (W ) ∼ = Z/2 and with non-trivial Kirby-Siebenmann invariant.…”

Isotopy of the Dehn twist on K3#K3 after a single stabilization

“…‚ Ruberman [28] gave the first example of self-diffeomorphisms on 4-manifolds that are isotopic to the identity in the topological category but not smoothly so. Further examples are given by Auckly-Kim-Melvin-Ruberman [4], Akbulut [3], Baraglia-Konno [7] and Kronheimer-Mrowka [22]. ‚ By the combined work of Wall [32], Perron [26], Quinn [27] and Donaldson [14],…”

“…Note that Kronheimer-Mrowka [22] proved that δ itself is not smoothly isotopic to the identity, using the nonequivariant Bauer-Furuta invariant for spin families. Our result is based on Kronheimer-Mrowka's theorem and makes use of the Pinp2q-equivariant version of the Bauer-Furuta invariant.…”

“…The idea of using gauge-theoretic invariant for families to study isotopy problem first appears in [28]. The idea of using Pinp2q-equivariant Bauer-Furuta invariant to further study Dehn twists on 4-manifolds was suggested by Kronheimer-Mrowka in [22].…”