A splitting theorem for the Seiberg-Witten invariant of a homology S1 × S3

Lin, Jianfeng¹ ; Ruberman, Daniel² ; Saveliev, Nikolai³

¹ Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, United States
² Department of Mathematics, Brandeis University, Waltham, MA, United States
³ Department of Mathematics, University of Miami, Coral Gables, FL, United States

Geometry & topology, Tome 22 (2018) no. 5, pp. 2865-2942.

Résumé

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

Détail

DOI : 10.2140/gt.2018.22.2865

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

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