Rohlin's invariant and gauge theory III. Homology 4-tori

Daniel Ruberman, Nikolai Saveliev

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

This is the third in our series of papers relating gauge theoretic invariants of certain 4-manifolds with invariants of 3-manifolds derived from Rohlin's theorem. Such relations are well-known in dimension three, starting with Casson's integral lift of the Rohlin invariant of a homology sphere. We consider two invariants of a spin 4-manifold that has the integral homology of a 4-torus. The first is a degree zero Donaldson invariant, counting flat connections on a certain SO(3)-bundle. The second, which depends on the choice of a 1-dimensional co-homology class, is a combination of Rohlin invariants of a 3-manifold carrying the dual homology class. We prove that these invariants, suitably normalized, agree modulo 2, by showing that they coincide with the quadruple cup product of 1-dimensional cohomology classes.

Original languageEnglish (US)
JournalGeometry and Topology
Volume9
StatePublished - Oct 27 2005

Fingerprint

Invariant Theory
Gauge Theory
Homology
Torus
Invariant
4-manifold
Cup Product
Homology Spheres
Flat Connection
Quadruple
Three-dimension
Modulo
Cohomology
Counting
Bundle
Gauge
Series
Zero
Theorem
Class

Keywords

  • Donaldson invariant
  • Equivariant perturbation
  • Homology torus
  • Rohlin invariant

ASJC Scopus subject areas

  • Geometry and Topology

Cite this

Rohlin's invariant and gauge theory III. Homology 4-tori. / Ruberman, Daniel; Saveliev, Nikolai.

In: Geometry and Topology, Vol. 9, 27.10.2005.

Research output: Contribution to journalArticle

@article{354236265e3f4186ae35c8311001b1ec,
title = "Rohlin's invariant and gauge theory III. Homology 4-tori",
abstract = "This is the third in our series of papers relating gauge theoretic invariants of certain 4-manifolds with invariants of 3-manifolds derived from Rohlin's theorem. Such relations are well-known in dimension three, starting with Casson's integral lift of the Rohlin invariant of a homology sphere. We consider two invariants of a spin 4-manifold that has the integral homology of a 4-torus. The first is a degree zero Donaldson invariant, counting flat connections on a certain SO(3)-bundle. The second, which depends on the choice of a 1-dimensional co-homology class, is a combination of Rohlin invariants of a 3-manifold carrying the dual homology class. We prove that these invariants, suitably normalized, agree modulo 2, by showing that they coincide with the quadruple cup product of 1-dimensional cohomology classes.",
keywords = "Donaldson invariant, Equivariant perturbation, Homology torus, Rohlin invariant",
author = "Daniel Ruberman and Nikolai Saveliev",
year = "2005",
month = "10",
day = "27",
language = "English (US)",
volume = "9",
journal = "Geometry and Topology",
issn = "1465-3060",
publisher = "University of Warwick",

}

TY - JOUR

T1 - Rohlin's invariant and gauge theory III. Homology 4-tori

AU - Ruberman, Daniel

AU - Saveliev, Nikolai

PY - 2005/10/27

Y1 - 2005/10/27

N2 - This is the third in our series of papers relating gauge theoretic invariants of certain 4-manifolds with invariants of 3-manifolds derived from Rohlin's theorem. Such relations are well-known in dimension three, starting with Casson's integral lift of the Rohlin invariant of a homology sphere. We consider two invariants of a spin 4-manifold that has the integral homology of a 4-torus. The first is a degree zero Donaldson invariant, counting flat connections on a certain SO(3)-bundle. The second, which depends on the choice of a 1-dimensional co-homology class, is a combination of Rohlin invariants of a 3-manifold carrying the dual homology class. We prove that these invariants, suitably normalized, agree modulo 2, by showing that they coincide with the quadruple cup product of 1-dimensional cohomology classes.

AB - This is the third in our series of papers relating gauge theoretic invariants of certain 4-manifolds with invariants of 3-manifolds derived from Rohlin's theorem. Such relations are well-known in dimension three, starting with Casson's integral lift of the Rohlin invariant of a homology sphere. We consider two invariants of a spin 4-manifold that has the integral homology of a 4-torus. The first is a degree zero Donaldson invariant, counting flat connections on a certain SO(3)-bundle. The second, which depends on the choice of a 1-dimensional co-homology class, is a combination of Rohlin invariants of a 3-manifold carrying the dual homology class. We prove that these invariants, suitably normalized, agree modulo 2, by showing that they coincide with the quadruple cup product of 1-dimensional cohomology classes.

KW - Donaldson invariant

KW - Equivariant perturbation

KW - Homology torus

KW - Rohlin invariant

UR - http://www.scopus.com/inward/record.url?scp=27644545389&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=27644545389&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:27644545389

VL - 9

JO - Geometry and Topology

JF - Geometry and Topology

SN - 1465-3060

ER -