Homology Chern characters of perturbed Dirac operators (Q2747327)

The paper addresses the problem of expressing topological invariants of complete noncompact Riemannian manifolds in terms of analytical invariants (index, higher indices) of perturbed Dirac-type operators.NEWLINENEWLINENEWLINEIt is well known (see for example the Bunke article cited in the paper) that some topological invariants of such manifolds can be expressed as indices of Dirac-type operators perturbed by a zero-order differential operator (a bundle map) provided the manifold corona is the Higson one. Bunke confines himself to perturbations, which are bounded at infinity. The Kasparov KK-theory provides a suitable link between the operators and the topology of the underlying manifold. Namely the perturbed operator defines an element of \(KK(B,C)\) for a subalgebra \(B\) of \(C_0(M)\), which satisfies certain technical condition. Computability of the invariants encoded in said element of \(KK(B,C)\) depends on expressing it as the Kasparov product of elements defined by the unperturbed Dirac operator and the perturbation.NEWLINENEWLINENEWLINEThe authors discuss Bunke results in the context of special kind of manifolds \(M\), namely those, which outside a compact are warped products \(R_{\geq 0}\times N\), where \(N\) is a Riemannian manifold with metric \(g_N\), and the metric on the warped product is of the form \(dr^2+h^2 (r)g_N\), \(r\) being the natural parameter on \(R_{\geq 0}\) and \(h(r)\) a positive function. The main conclusion of the discussion is that the Bunke results apply only to expanding ends \((h(r)\to \infty)\), or at least to ends, on which \(h(r)\) is separated from zero, and this restriction is attributed to the property of the perturbation of being bounded. In the case of cusps for example (and bounded perturbations), the above-mentioned technical condition on the subalgebra \(B\) makes it to ``small'' to reflect topological structure of the manifold's end (i.e. the manifold \(N)\).NEWLINENEWLINENEWLINEIn order to extend Bunke results to the case of coronas larger than the Higson one and to collapsing ends, the authors consider thoroughly unbounded perturbations of Dirac type operators i.e. perturbations the infimum of the spectrum of which tends to infinity whenever the point \(x\in M\) tends to infinity. The authors prove, and this is the main result of the paper, that under certain additional technical conditions, such perturbed Dirac-type operators define elements of \(KK(B,C)\), which are Kasparov products of the elements corresponding to the unperturbed Dirac operator and the perturbation. Moreover the authors indicate that the algebra \(B\) can be large enough to reflect the topology of the collapsing end for suitably chosen thoroughly unbounded perturbation.