G-monopole invariants on some connected sums of 4-manifolds

TL;DR

This paper introduces and computes G-monopole invariants on certain 4-manifolds with group actions, revealing nontrivial invariants for connected sums with cyclic symmetry, advancing understanding of equivariant Seiberg-Witten theory.

Contribution

It defines G-monopole invariants for 4-manifolds with finite group actions and computes these invariants for specific connected sums, demonstrating nontrivial cases.

Findings

G-monopole invariants are nonzero for certain connected sums with cyclic symmetry.

The invariants detect nontrivial Seiberg-Witten structures in equivariant settings.

Connected sums of manifolds with nontrivial mod 2 invariants have nonzero cyclic group invariants.

Abstract

On a smooth closed oriented $4$-manifold $M$ with a smooth action of a finite group $G$ on a Spin$^c$ structure, $G$-monopole invariant is defined by "counting" $G$-invariant solutions of Seiberg-Witten equations for any $G$-invariant Riemannian metric on $M$. We compute $G$-monopole invariants on some $G$-manifolds. For example, the connected sum of $k$ copies of a 4-manifold with nontrivial mod 2 Seiberg-Witten invariant has nonzero $\Bbb Z_k$-monopole invariant mod 2, where the $\Bbb Z_k$-action is given by cyclic permutations of $k$ summands.