The signature package on Witt spaces, I. Index classes

Details The signature package on Witt spaces, I. Index classes The signature package on Witt spaces, I. Index classes

2009-06-08

arxiv.org/abs/0906.1568v2

Mathematics

Differential Geometry

small changes to agree with part II

Scientific paper

We give a parametrix construction for the signature operator on any compact, oriented, stratified pseudomanifold X which satisfies the Witt condition. This construction is inductive. It is then used to show that the signature operator is essentially self-adjoint and has discrete spectrum of finite multiplicity, so that its index -- the analytic signature of X -- is well-defined. We then show how to couple this construction to a C^*_r(Gamma) Mischenko bundle associated to any Galois covering of X with covering group Gamma. The appropriate analogues of these same results are then proved, and it follows that we may define an analytic signature class as an element of the K-theory of C^*_r(Gamma). In a sequel to this paper we establish in this setting the full range of conclusions for this class which sometimes goes by the name of the signature package.

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