## Abstract

We study the Seiberg-Witten invariant λ_{SW}(X) of smooth spin 4 -manifolds X with the rational homology of S^{1} × S^{3} defined by Mrowka, Ruberman and Saveliev as a signed count of irreducible monopoles amended by an index-theoretic correction term. We prove a splitting formula for this invariant in terms of the Frøyshov invariant h(X) and a certain Lefschetz number in the reduced monopole Floer homology of Kronheimer and Mrowka. We apply this formula to obstruct the existence of metrics of positive scalar curvature on certain 4 -manifolds, and to exhibit new classes of homology 3 -spheres of infinite order in the homology cobordism group.

Original language | English (US) |
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Pages (from-to) | 2865-2942 |

Number of pages | 78 |

Journal | Geometry and Topology |

Volume | 22 |

Issue number | 5 |

DOIs | |

State | Published - Jun 1 2018 |

## Keywords

- Frøyshov invariant
- Manifolds with periodic ends
- Monopole floer homology
- Seiberg-witten theory

## ASJC Scopus subject areas

- Geometry and Topology