A splitting theorem for the Seiberg-Witten invariant of a homology $S^1\times S^3$

Abstract

We study the Seiberg-Witten invariant ${\lambda }_{\mathrm{SW}}(X)$ of smooth spin $4$-manifolds $X$ with integral homology of $S^1\times S^3$ 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 existence of metrics of positive scalar curvature on certain 4-manifolds, and to exhibit new classes of integral homology $3$-spheres of Rohlin invariant one which have infinite order in the homology cobordism group.

78 pages, 1 figure; added results on Z/2 homology cobordisms and knot concordances

References in corpus (7)

• Floer homology and knot complements
• On the Khovanov and knot Floer homologies of quasi-alternating links
• Involutive Heegaard Floer homology
• Pin(2)-equivariant Seiberg-Witten Floer homology of Seifert fibrations
• Manolescu Invariants of Connected Sums
• Dirac operators on manifolds with periodic ends
• A Morse-Bott approach to monopole Floer homology and the Triangulation conjecture

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• A Generalization of the Tristram-Levine Knot Signatures as a Singular Furuta-Ohta Invariant for Tori
• The Seiberg-Witten equations on end-periodic manifolds and an obstruction to positive scalar curvature metrics
• 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
• Seifert hypersurfaces of 2-knots and Chern-Simons functional
• A Splitting Formula in Instanton Floer Homology
• On the spectral sets of Inoue surfaces
• Seiberg-Witten-Casson Invariant of Homology $S^1\times S^3$ with Circle Action