A dimensionally reduced Gysin sequence for the equivariant Brauer group

Abstract

In 2005 Bouwknegt, Hannabuss and Mathai constructed a "dimensionally reduced" Gysin sequence for de Rham cohomology using a Chern-Weil dimensional reduction isomorphism. This sequence was then used to propose a T-dual curvature class of a principal torus bundle equipped with an integral 3 cohomology class, called a H-flux. However, being a sequence for de Rham cohomology, it invariably omitted the phenomena of torsion. Somewhat earlier, in 1996, Packer, Raeburn and Williams, using the theory of group actions on continuous trace algebras, had written an integer cohomology Gysin sequence, that therefore includes torsion. However, this sequence is only able to compute the T-dual curvature class in the case where the projection of the H-flux to the E {u00B9} {u00B2}{u221E} term of the Leray-Serre spectral sequence is trivial. We generalise both of these results by constructing an integer cohomology Gysin sequence that applies to the case where the projection of the H-flux to the E {u2070} {u00B3}{u221E} term of the Leray-Serre spectral sequence is trivial. This subgroup is significant, as it describes precisely those H-fluxes that can be lifted to an element of the equivariant Brauer group. The construction utilises the groupoid cohomology of Tu and dimensional reduction isomorphisms for the equivariant Brauer group and integer {u010C}ech cohomology. From this point of view, the forgetful homomorphism appears as a connecting homomorphism in a long exact sequence.