Periodic cyclic cohomology Chern character for pseudomanifolds with one singular stratum

We compute the periodic cyclic cohomology Chern character of an admissible pseudomanifold Xf with one singular stratum. As a corollary, we obtain the index theorem and spectral flow for signature operators. 0. INTRODUCTION This paper is the sequel to [Chan]. In that paper, by considering the "straight" Chern character, we gave a de-Rham type realization of the Goresky-MacPhersonSiegel L-classes of admissible pseudomanifolds with one singular stratum Xt M U (ct (L) x N) such that 2L has zero oriented cobordism, and we also obtained the index theorem for the twisted signature operators on Xt. In this paper, we will continue to study Xt by using non-commutative geometry. We will compute (Theorem 3.1 and Theorem 3.3) the corresponding periodic cyclic cohomology Chern character by using the infinite temperature limit formula in [CoM]. As in [Chan], we will choose a scaling in the conical direction such that the signature operator is essentially self-adjoint. Also, by using a singular elliptic estimate, we handle the calculation on the singular part by that on model space c0,0, (L) x N. We finish the computation by Getzler's calculus. As a consequence, we recover the index theorem for twisted signature operators on Xt [Chan, Theorem 4.1] in the even case. In the odd case, we obtain (Corollary 3.4) the spectral flow for signature operators on admissible spaces with conical singularity. 1. PRELIMINARIES To fix the notation, let us recall some definitions in [Chan]. Let M be a smooth, oriented, compact and connected m-dimensional manifold with boundary O9M = L x N where L and N are smooth, oriented, closed and connected manifolds of dimensions ? and n respectively. Let c(L) -= (0, 1) x L and ct (L) = [0, 1) x L/{O} x L be the cone and completed cone with link L respectively. Then Xt = MU (ct (L) x N) is called a pseudomanifold with one singular stratum [Chan]. We define a metric g on Xt such that (i) g is a measurable metric on Xt; (ii) glM is a smooth metric on M and is a product near OM; Received by the editors August 30, 1996. 1991 Mathematics Subject Classification. Primary 19D55; Secondary 58G12. ?1998 American Mathematical Society 669 This content downloaded from 157.55.39.127 on Wed, 29 Jun 2016 04:18:19 UTC All use subject to http://about.jstor.org/terms