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

Daniel Ruberman, Nikolai Saveliev

Research output: Contribution to journalArticlepeer-review

12 Scopus citations

Abstract

This is the first in a series of papers exploring the relationship between the Rohlin invariant and gauge theory. We discuss a Casson-type invariant of a 3-manifold Y with the integral homology of the 3-torus, given by counting projectively flat U(2)-connections. We show that its mod 2 evaluation is given by the triple cup product in cohomology, and so it coincides with a certain sum of Rohlin invariants of Y. Our counting argument makes use of a natural action of H1(Y; ℤ2) 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.

Original languageEnglish (US)
Pages (from-to)618-646
Number of pages29
JournalCommentarii Mathematici Helvetici
Volume79
Issue number3
DOIs
StatePublished - Jan 1 2004

Keywords

  • Casson invariant
  • Flat moduli spaces
  • Floer homology
  • Rohlin invariant

ASJC Scopus subject areas

  • Mathematics(all)

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