Finite group actions and G-monopole classes on smooth 4-manifolds

Details Finite group actions and G-monopole classes on smooth 4-manifolds Finite group actions and G-monopole classes on smooth 4-manifolds

2011-08-19

arxiv.org/abs/1108.3875v1

Mathematics

Geometric Topology

Scientific paper

On a smooth closed oriented 4-manifold M with a smooth action by a compact Lie group G, we define a G-monopole class as an element of $H^2(M;\Bbb Z)$ which is the first Chern class of a G-equivariant Spin$^c$ structure which has a solution of the Seiberg-Witten equations for any G-invariant Riemannian metric on M. We find $\Bbb Z_k$-monopole classes on some $\Bbb Z_k$-manifolds such as the connected sum of k copies of a 4-manifold with nontrivial mod 2 Seiberg-Witten invariant or Bauer-Furuta invariant, where the $\Bbb Z_k$-action is a cyclic permutation of k summands. As an application, we produce infinitely many exotic non-free actions of $\Bbb Z_k\oplus H$ on some connected sums of finite number of $S^2\times S^2$, $\Bbb CP_2$, $\bar{\Bbb CP}_2$, and K3 surfaces, where $k\geq 2$, and H is any finite group acting freely on $S^3$.

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