3 - Fold Local index theorem

Although this is a slightly modified version of the paper [23], it has to be seen as preliminary work.3-Fold Local Index Theorem means Local (Local (Local Index Theorem))). Local Index Theorem is the Connes-Moscovici local index theorem [4], [5]. The second ’’Local” refers to the cyclic homology localised to a certain separable subring of the ground algebra, while the last one refers to Alexander-Spanier type cyclic homology. Localised cyclic homology had already appeared in the literature, see Connes [3], Karoubi [9] [10], Loday [12].The Connes-Moscovici work is based on the operator R.(A) = P - e associated to the elliptic pseudo-differential operator A on the smooth manifold M, where P . e are idempotents, see [4], Pg. 353.The operator R(A) has two main merits: it is a smoothing operator and its distributional kernel is situated in an arbitrarily small neigh- bourhood of the diagonal in M x M.The operator R(A) has also two setbacks: -i) it is not an idempotent and therefore it does not have a genuine Connes-Karoubi-Chern character in the absolute cyclic homology of the algebra of smoothing operators, see Connes [2], [3], Karoubi [9] [10] ; -ii) even if it were an idempotent, its Connes-Karoubi-Chern character would belong to the cyclic homology of the algebra of smoothing operators with arbitrary supports, which is trivial.This paper presents a new solution to the difficulties raised by the two setbacks.For which concerns -i), we show that although R(A) is not an idem- potent, it satisfies the identity (R(A))2 = R(-A) - [R(A).e + e.R(A)]. We show that the operator R(A) has a genuine Chern character provided the cyclic homology complex of the algebra of smoothing operators is localised to the separable sub-algebra A = C + C.e, see Section 7.1.For which concerns -ii), we introduce the notion of local cyclic homology; this is constructed on the foot-steps of the Alexander-Spanier homology, i.e. by filtering the chains of the cyclic homology complex of the algebra of smoothing operators by their distributional support, see Section 6.Using these new instruments, we give a reformulation of the Connes- Moscovici local Index Theorem, see Theorem 8.1, Section 8. As a corollary of this theorem, we show that the local cyclic homology of the algebra of smoothing operators is at least as big as the Alexander-Spanier homology of the base manifold.The present reformulation of Connes-Moscovici local index theorem opens the way to new investigations, see Section 9.