On the Frøyshov invariant and monopole Lefschetz number

Abstract

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.

arXiv comment77 pages, 7 figures

References in corpus (13)

• On the Floer homology of plumbed three-manifolds
• On the Khovanov and knot Floer homologies of quasi-alternating links
• HF=HM I : Heegaard Floer homology and Seiberg--Witten Floer homology
• Contact structures, sutured Floer homology and TQFT
• The equivalence of Heegaard Floer homology and embedded contact homology via open book decompositions II
• HF=HM IV: The Seiberg-Witten Floer homology and ech correspondence
• HF=HM V: Seiberg-Witten-Floer homology and handle addition
• Stable isotopy in four dimensions
• Surgery formula for the renormalized Euler characteristic of Heegaard Floer homology
• Concordance groups of links
• An absolute grading on Heegaard Floer homology by homotopy classes of oriented 2-plane fields
• Dirac operators on manifolds with periodic ends
• The theta divisor and the Casson-Walker invariant

Cited by in corpus (9)

• On the monopole Lefschetz number of finite order diffeomorphisms
• Gluing maps and cobordism maps for sutured monopole Floer homology
• A Generalization of the Tristram-Levine Knot Signatures as a Singular Furuta-Ohta Invariant for Tori
• Instanton Floer homology, sutures, and Euler characteristics
• Rank inequalities for the Heegaard Floer homology of branched covers
• A Surgery Formula for the Casson-Seiberg-Witten Invariant of Integral Homology $S^1\times S^3$
• Positive scalar curvature and $10/8$-type inequalities on $4$-manifolds with periodic ends
• Seiberg-Witten Floer homotopy contact invariant
• Positive scalar curvature and homology cobordism invariants