Equivariant Bauer-Furuta invariants on Some Connected Sums of 4-manifolds

Abstract

On some connected sums of 4-manifolds with natural actions of finite groups, we use equivariant Bauer-Furuta invariant to deduce the existence of solutions of Seiberg-Witten equations invariant under the group actions. For example, for any integer $k\geq 2$ we show that the connected sum of $k$ copies of a 4-manifold $M$ with nontrivial Bauer-Furuta invariant has a nontrivial $\mathbb{Z}_k$-equivariant Bauer-Furuta invariant for the obviously glued Spin$^c$ structure, where the $\mathbb{Z}_k$-action cyclically permutes $k$ summands of $M$. This contrasts with the fact that ordinary Bauer-Furuta invariants of such connected sums are all trivial for any sufficiently large $k$, when $b_1(M)=0$.