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

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

2003-02-11

arxiv.org/abs/math/0302131v2

Comm. Math. Helv., 79 (2004), no. 3, 618--646.

Mathematics

Geometric Topology

Changed title to fit with succeeding papers in series. Added reference to Turaev's work

Scientific paper

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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