On the Frøyshov invariant and monopole Lefschetz number

林剑锋 清华大学丘成桐数学科学中心 Daniel Ruberman Brandeis University Nikolai Saveliev University of Miami

Geometric Analysis and Geometric Topology mathscidoc:2205.15006

Journal of Differential Geometry, 2021.7
Given an involution on a rational homology 3-sphere Y with quotient the 3-sphere, we prove a formula for the Lefschetz number of the map induced by this involution in the reduced monopole Floer homology. This formula is motivated by a variant of Witten's conjecture relating the Donaldson and Seiberg--Witten invariants of 4-manifolds. A key ingredient is a skein-theoretic argument, making use of an exact triangle in monopole Floer homology, that computes the Lefschetz number in terms of the Murasugi signature of the branch set and the sum of Frøyshov invariants associated to spin structures on Y. We discuss various applications of our formula in gauge theory, knot theory, contact geometry, and 4-dimensional topology.
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  • Accepted by Journal of Differential Geometry
  title={On the Frøyshov invariant and monopole Lefschetz number},
  author={林剑锋, Daniel Ruberman, and Nikolai Saveliev},
  booktitle={Journal of Differential Geometry},
林剑锋, Daniel Ruberman, and Nikolai Saveliev. On the Frøyshov invariant and monopole Lefschetz number. 2021. In Journal of Differential Geometry. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220517175110608647251.
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