Rohlin's invariant and gauge theory, I. Homology 3-tori

Daniel Ruberman; Nikolai Saveliev

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arXiv · math/0302131

Rohlin's invariant and gauge theory, I. Homology 3-tori

Daniel Ruberman·Nikolai Saveliev

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Abstract

This is the first in a series of papers exploring the relationship between the Rohlin invariant and gauge theory. We discuss the Casson-type invariant of a 3-manifold with the integral homology of a torus, given by counting projectively flat connections. We show that its mod 2 evaluation is given by the triple cup product in cohomology, and so it coincides with a sum of Rohlin invariants. Our counting argument makes use of a natural action of the first cohomology on the moduli space of projectively flat connections; along the way we construct perturbations that are equivariant with respect to this action. Combined with the Floer exact triangle, this gives a purely gauge-theoretic proof that Casson's homology sphere invariant reduces mod 2 to the Rohlin invariant.

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BibTeX RIS

Daniel Ruberman, Nikolai Saveliev. 2003-06-09. Rohlin's invariant and gauge theory, I. Homology 3-tori. https://arxiv.org/abs/math/0302131

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