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 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 | |
| State | Published - Jun 1 2018 |
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Keywords
- Frøyshov invariant
- Manifolds with periodic ends
- Monopole floer homology
- Seiberg-witten theory
ASJC Scopus subject areas
- Geometry and Topology
Cite this
A splitting theorem for the seiberg-witten invariant of a homology S1 × S3 . / Lin, Jianfeng; Ruberman, Daniel; Saveliev, Nikolai.
In: Geometry and Topology, Vol. 22, No. 5, 01.06.2018, p. 2865-2942.Research output: Contribution to journal › Article
}
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
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 -