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

• Autores: Daniel Ruberman, Nikolai Saveliev
• Localización: Commentarii mathematici helvetici, ISSN 0010-2571, Vol. 79, Nº 3, 2004, págs. 618-646
• Idioma: inglés
• DOI: 10.1007/s00014-004-0816-y
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• Resumen

  • 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 H^1 (Y;Z_2) 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