ar X iv : 0 90 6 . 15 68 v 1 [ m at h . D G ] 8 J un 2 00 9 THE SIGNATURE PACKAGE ON WITT SPACES , I . INDEX CLASSES

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 rΓ Mischenko bundle associated to any Galois covering of X with covering group Γ. 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 rΓ. 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.