Bordism invariance of the coarse index (Q2845543)

The author introduces the notion of a directed \(c\)-bordism, in which the boundary of a smooth manifold \(W\) decomposes into smooth manifolds \(N_{1}\) and \(N_{2}\), with the inclusion of \(N_{1}\) into \(W\) a coarse map and the inclusion of \(N_{2}\) into \(W\) a coarse equivalence. This notion of directed bordism extends naturally to directed bordism of operators \(D_{1}\) (on \(N_{1}\)) and \(D_{2}\) (on \(N_{2}\)) that are of Dirac type or more generally are elliptic pseudodifferential operators that define \(K\)-homology classes. The author shows that a directed \(c\)-bordism induces a map, from the \(K\)-theory of \(N_{1}\)'s coarse \(C^{*}\)-algebra to the \(K\)-theory of \(N_{2}\)'s coarse \(C^{*}\)-algebra, that takes the coarse index of \(D_{1}\) to the coarse index of \(D_{2}\). The coarse index of a spin manifold's Dirac operator is an obstruction to the existence of a Riemannian metric with uniformly positive scalar curvature.NEWLINENEWLINEDirected \(c\)-bordisms extend coarse index theory's implications for the non-existence of metrics of uniformly positive scalar curvature. When \(W\) is the product of an interval with a manifold \(N\), a pair of complete quasi-isometric Riemannian metrics \(g_{0}\) and \(g_{1}\) on \(N\) can be used to determine \(c\)-bordisms that run in either direction between \((N,g_{0})\) and \((N,g_{1})\). The author states an inequality involving path metrics that generalizes quasi-isometry but still defines a \(c\)-bordism in one direction. Hence the author can strengthen a result in [\textit{B. Hanke} et al., Ann. Sci. Éc. Norm. Supér. (4) 41, No. 3, 473--495 (2008; Zbl 1169.53032)] that a spin universal cover of a uniformly enlargeable manifold \(M\) does not admit a Riemannian metric of uniformly positive scalar curvature that is quasi-isometric to a metric pulled back from \(M\). In the new result, quasi-isometry is replaced by the weaker inequality between path metrics, \(d_{0} \geq C d_{1} - K\), where \(C\) and \(K\) are positive, \(d_{1}\) represents the path metric associated with the Riemannian metric pulled back from \(M\), and \(d_{0}\) represents the path metric associated with the Riemannian metric that is being shown not to have uniformly positive scalar curvature.