A splitting theorem for the seiberg-witten invariant of a homology S1 × S3

Jianfeng Lin; Daniel Ruberman; Nikolai Saveliev

doi:10.2140/gt.2018.22.2865

A splitting theorem for the seiberg-witten invariant of a homology S¹ × S³

Jianfeng Lin, Daniel Ruberman, Nikolai Saveliev

Mathematics

Research output: Contribution to journal › Article › peer-review

3 Scopus citations

Abstract

We study the Seiberg-Witten invariant λ_SW(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.

Original language	English (US)
Pages (from-to)	2865-2942
Number of pages	78
Journal	Geometry and Topology
Volume	22
Issue number	5
DOIs	https://doi.org/10.2140/gt.2018.22.2865
State	Published - Jun 1 2018

Keywords

Frøyshov invariant
Manifolds with periodic ends
Monopole floer homology
Seiberg-witten theory

ASJC Scopus subject areas

Geometry and Topology

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title = "A splitting theorem for the seiberg-witten invariant of a homology S1 × S3",

abstract = "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 Fr{\o}yshov 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.",

keywords = "Fr{\o}yshov invariant, Manifolds with periodic ends, Monopole floer homology, Seiberg-witten theory",

author = "Jianfeng Lin and Daniel Ruberman and Nikolai Saveliev",

note = "Funding Information: We are thankful to Tye Lidman, Ciprian Manolescu and Matthew Stoffregen for generously sharing their expertise. Lin was partially supported by NSF grant DMS-1707857, Ruberman was partially supported by NSF grant DMS-1506328 and Saveliev was partially supported by a Collaboration Grant from the Simons Foundation. Publisher Copyright: {\textcopyright} 2018, Mathematical Sciences Publishers. All Rights reserved.",

year = "2018",

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doi = "10.2140/gt.2018.22.2865",

language = "English (US)",

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T1 - A splitting theorem for the seiberg-witten invariant of a homology S1 × S3

AU - Lin, Jianfeng

AU - Ruberman, Daniel

AU - Saveliev, Nikolai

N1 - Funding Information: We are thankful to Tye Lidman, Ciprian Manolescu and Matthew Stoffregen for generously sharing their expertise. Lin was partially supported by NSF grant DMS-1707857, Ruberman was partially supported by NSF grant DMS-1506328 and Saveliev was partially supported by a Collaboration Grant from the Simons Foundation. Publisher Copyright: © 2018, Mathematical Sciences Publishers. All Rights reserved.

PY - 2018/6/1

Y1 - 2018/6/1

N2 - 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.

AB - 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.

KW - Frøyshov invariant

KW - Manifolds with periodic ends

KW - Monopole floer homology

KW - Seiberg-witten theory

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