## Abstract

Let K be a knot in an integral homology 3–sphere Y and † the corresponding n–fold cyclic branched cover. Assuming that † is a rational homology sphere (which is always the case when n is a prime power), we give a formula for the Lefschetz number of the action that the covering translation induces on the reduced monopole homology of †. The proof relies on a careful analysis of the Seiberg–Witten equations on 3–orbifolds and of various n–invariants. We give several applications of our formula: (1) we calculate the Seiberg–Witten and Furuta–Ohta invariants for the mapping tori of all semifree actions of Z=n on integral homology 3–spheres; (2) we give a novel obstruction (in terms of the Jones polynomial) for the branched cover of a knot in S^{3} being an L–space; and (3) we give a new set of knot concordance invariants in terms of the monopole Lefschetz numbers of covering translations on the branched covers.

Original language | English (US) |
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Pages (from-to) | 3591-3628 |

Number of pages | 38 |

Journal | Geometry and Topology |

Volume | 25 |

Issue number | 7 |

DOIs | |

State | Published - 2021 |

Externally published | Yes |

## ASJC Scopus subject areas

- Geometry and Topology