#### Volume 22, issue 5 (2018)

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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
##### Abstract

We study the Seiberg–Witten invariant ${\lambda }_{SW}\left(X\right)$ of smooth spin $4$–manifolds $X$ with the rational homology of ${S}^{1}×{S}^{3}$ 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 $h\left(X\right)$ 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.

##### Keywords
Seiberg–Witten theory, monopole Floer homology, Frøyshov invariant, manifolds with periodic ends
##### Mathematical Subject Classification 2010
Primary: 57R57, 57R58
Secondary: 53C21, 57M27, 58J28