A \(K\)-theory proof of the cobordism invariance of the index (Q864959)

The cobordism invariance of the index is a generalization of \textit{R. Thom}'s remark [''Quelques propriétés des variétés-bords'', Colloque de Topologie de Strasbourg 1951, No. 5 (1952; Zbl 0053.30102)] on the vanishing of the topological signature vanishing for boundaries of compact manifolds. The cobordism invariance of twisted signatures was an essential ingredient in the first proof of the Atiyah-Singer index theorem, cf.\ [\textit{R. S. Palais}, Seminar on the Atiyah-Singer index theorem. Princeton University Press. (1965; Zbl 0137.17002)]. It also follows from the Atiyah-Singer formula that the index of spin Dirac operators vanishes on manifolds which bound. The main result of the present paper is a cobordism invariance theorem for general elliptic pseudodifferential operators which are multiplication at infinity. The statement involves \(K\)-theory. Let \(M\) be a \(\sigma\)-compact manifold, and \(\alpha\) an elliptic symbol on \(M\) which is multiplication outside a compact set (the index is known to depend only on the homotopy class of the symbol). Assume that \(M\) is the boundary of a \(\sigma\)-compact manifold \(X\), and that the \(K\)-theory class \([\alpha]\in K^0(TM)\) lives in the image of the composite map \(K^1(TX)\to K^1(TX_{| M})\to K^0(TM)\), obtained by restricting to \(M\) and then by aplying Bott periodicity. Then the index of \(\alpha\) is \(0\). To the knowledge of this reviewer, even in the compact case the above statement could not be found in literature, although informal (and sometimes incorrect) statements about cobordism invariance of the index seem to be folklore. The proof relies on the properties of the push-forward map in \(K\)-theory introduced by \textit{M. F. Atiyah} and \textit{I. M. Singer} in their second proof of the index theorem [Ann. Math. (2) 87, 484--530 (1968; Zbl 0164.24001)].