Geometry and Topology Seminar

Kronheimer and Mrowka showed that the Dehn twist along a $3$-sphere in the neck of the connected sum of two $K3$ surfaces is not smoothly isotopic to the identity. The tool they used is the nonequivariant family Bauer-Furuta invariant, and their result requires that the manifolds are simply connected and the signature of one of them is $16 \mod 32$. We generalize the $S^1$-equivariant family Bauer-Furuta invariant to nonsimply connected manifolds, and construct a refinement of this invariant. We use it to show that if $X_1,X_2$ are two homology tori whose determinants $r_1,r_2$ are odd, then the Dehn twist along a $3$-sphere in the neck of $X_1\mathbin{\#} X_2$ is not smoothly isotopic to the identity.