Cobordism invariance of the family index (Q2888900)

In the previous author's paper ``A \(K\)-theory proof of the cobordism invariance of the index'' [\(K\)-Theory 36, No. 1--2, 1--31 (2005; Zbl 1117.58009)], a proof of cobordism invariance that applies to general elliptic pseudodifferential operator, under suitable conditions on their \(K\)-theory symbol classes, defining so called symbol-cobordism, is given (see also \textit{S. Moroianu} [``On Carvalho's proof of the cobordism invariance of the index'', Proc. Am. Math. Soc. 134, No. 11, 3395--3404 (2006; Zbl 1111.58020)], for an analytic formulation of this result).NEWLINENEWLINEIn this paper, a \(K\)-theory proof of this cobordism invariance for families of elliptic pseudodifferential operators on closed manifolds is given. The main result is the following:NEWLINENEWLINETheorem: Let \(P\) be an elliptic family of pseudodifferential operators on a manifold \({\mathcal M}\) over \(B\), with symbol \(\sigma(P)\in K^0(T{\mathcal M})\). If \(\chi\) is such that \(\partial_B\chi={\mathcal M}\) and \(\sigma(P)\in\text{{Im}}u_{\chi}\), then \(\text{{ind}}(P)=0\).NEWLINENEWLINEThe principal point in the proof is to use push-forward maps and functoriality of the family index in \(K\)-theory. In particular, the author establishes the condition on the symbol of a given elliptic pseudodifferential family on a boundary that yield the vanishing of its index in the \(K\)-theory of the base.