4 A. L. CAREY, V. GAYRAL, A. RENNIE, and F. A. SUKOCHEV

For the application to noncompact manifolds M, we find that our noncommu-

tative index theorem dictates that the appropriate algebra A consists of smooth

functions which, together with all their derivatives, lie in

L1(M).

We show how

to construct K-homology classes for this algebra from the Dirac operator on the

spinor bundle over M. This K-homology viewpoint is related to Roe’s approach

[52] and to the relative index theory of [29].

Then the results, for Dirac operators coupled to connections on sections of bun-

dles over noncompact manifolds of bounded geometry, essentially follow as corol-

laries of the work of Ponge [47]. The theorems we obtain for even dimensional

manifolds are not comparable with those in [51], but are closely related to the

viewpoint of Gromov-Lawson [29]. For odd dimensional manifolds we obtain an

index theorem for generalised Toeplitz operators that appears to be new, although

one can see an analogy with the results of H¨ ormander [34, section 19.3].

We now digress to give more detail on how, for noncompact even dimensional

spin manifolds M, our local index formula implies a result analogous to the Gromov-

Lawson relative index theorem [29]. What we compute is an index pairing of K-

homology classes for the algebra A of smooth functions which, along with their

derivatives, all lie in

L1(M),

with differences of classes [E] − [E ] in the K-theory

of A. We verify that the Dirac operator on a spin manifold of bounded geometry

satisfies the hypotheses needed to use our residue cocycle formula so that we obtain

a local index formula of the form

(0.1) [E] − [E ], [D] = (const) A(M)(Ch(E) − Ch(E )),

where Ch(E) and Ch(E ) are the Chern classes of vector bundles E and E over

M. We emphasise that in our approach, the connections that lead to the curvature

terms in Ch(E) and Ch(E ), do not have to coincide outside a compact set as in

[29]. Instead they satisfy constraints that make the difference of curvature terms

integrable over M.

We reiterate that, for our notion of spectral triple, the operator D need not be

Fredholm and that the choice of the algebra A is dictated by the noncommutative

theory developed in Chapter 2. In that chapter we explain the minimal assumptions

on the pair (D, A) such that we can define a Kasparov module and so a KK-class.

The further assumptions required for the local index formula are specified, almost

uniquely, by the noncommutative integration theory developed in Chapter 1. We

verify (in Chapter 4) what these assumptions mean for the commutative algebra A

of functions on a manifold and Dirac-type operator D, in the case of a noncompact

manifold of bounded geometry, and prove that in this case we do indeed obtain a

spectral triple in the sense of our general definition.

In the odd dimensional case, for manifolds of bounded geometry, we obtain an

index formula that is apparently new, although it is of APS-type. The residues in

the noncommutative formula are again calculable by the techniques employed by

[47] in the compact case. This results in a formula for the pairing of the Chern

character of a unitary u in a matrix algebra over A, representing an odd K-theory

class, with the K-homology class of a Dirac-type operator D of the form

(0.2) [u], [D] = (const)

ˆ(M)Ch(u).

A

We emphasise that the assumptions on the algebra A of functions on M are such

that this integral exists but they do not require compact support conditions.