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

5 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

Access to Document

10.2140/gt.2018.22.2865

Cite this

@article{a9cde8731108479cba4a64c0ec86a1b9,

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",

month = jun,

day = "1",

doi = "10.2140/gt.2018.22.2865",

language = "English (US)",

volume = "22",

pages = "2865--2942",

journal = "Geometry and Topology",

issn = "1465-3060",

publisher = "University of Warwick",

number = "5",

}

TY - JOUR

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

UR - http://www.scopus.com/inward/record.url?scp=85048306379&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85048306379&partnerID=8YFLogxK

U2 - 10.2140/gt.2018.22.2865

DO - 10.2140/gt.2018.22.2865

M3 - Article

AN - SCOPUS:85048306379

VL - 22

SP - 2865

EP - 2942

JO - Geometry and Topology

JF - Geometry and Topology

SN - 1465-3060

IS - 5

ER -