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

Jianfeng Lin, Daniel Ruberman and Nikolai Saveliev

Geometry & Topology 22 (2018) 2865–2942

We study the Seiberg–Witten invariant λSW(X) of smooth spin 4–manifolds X with the rational homology of S1 × S3 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.

Seiberg–Witten theory, monopole Floer homology, Frøyshov invariant, manifolds with periodic ends
Mathematical Subject Classification 2010
Primary: 57R57, 57R58
Secondary: 53C21, 57M27, 58J28
Received: 7 March 2017
Accepted: 4 March 2018
Published: 1 June 2018
Proposed: Simon Donaldson
Seconded: András I Stipsicz, Ciprian Manolescu
Jianfeng Lin
Department of Mathematics
Massachusetts Institute of Technology
Cambridge, MA
United States
Daniel Ruberman
Department of Mathematics
Brandeis University
Waltham, MA
United States
Nikolai Saveliev
Department of Mathematics
University of Miami
Coral Gables, FL
United States