The odd Chern character and index localization formulae

We describe geometric representatives for the generatorsof the cohomology ring of a model of the classifying spacefor the functor $K^{-1}$. The class corresponding to thedegree one generator is closely related to the spectralflow of a one-parameter family of self-adjoint, Fredholmoperators. We use intersection theory to derivelocalization formulae that express the cohomological indexof a higher dimensional family of such operators as thePoincare dual of an explicit $0$-cycle in the parameterspace. We derive, under certain conditions, an equalitythat relates the cohomological index to the variation ofthe family of kernels.

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Published 2 September 2011