### 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 language | English (US) |
---|---|

Journal | Geometry and Topology |

Volume | 9 |

State | Published - Oct 27 2005 |

### Fingerprint

### Keywords

- Donaldson invariant
- Equivariant perturbation
- Homology torus
- Rohlin invariant

### ASJC Scopus subject areas

- Geometry and Topology

### Cite this

*Geometry and Topology*,

*9*.

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

Research output: Contribution to journal › Article

*Geometry and Topology*, vol. 9.

}

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 -