On vanishing of low-degree higher \(\hat{A}\)-genera
In this paper, the main result being a vanishing theorem as an implication of the strong Novikov conjecture, it is shown that for \(G\) a discrete group and \(BG\) its classifying space, for \(M\) a closed spin manifold with a Riemannian metric of positive scalar curvature, and for any element \(x\) of the subring of the rational homology theory of \(BG\) generated by elements of degree 1 or 2, the pairing between the cup product of the rational assembly map \(A\) related to the image \(\hat{A}(TM)\) of the tangent bundle \(TM\) of \(M\) and the pull back bundle of \(x\) by a continuous map from \(M\) to \(BG\), and the fundamental homology class of \(M\) is equal to zero. The proof involves the index formula and related computation for the Mishchenko-Fomenko index of the spin Dirac operator on \(M\) twisted by a Hilbert \(C^*\)-algebra module bundle constructed from a bundle of Hilbert spaces over \(M\) and a connection on this bundle. In the same situation as above, the strong Novikov conjecture is proved by \textit{B. Hanke} and \textit{T. Schick} [J. Differ. Geom. 74, No. 2, 293--320 (2006; Zbl 1122.58011); \(K\)-Theory 38, No. 1, 23--33 (2007; Zbl 1128.58012); Geom. Dedicata 135, 119--127 (2008; Zbl 1149.19006)], so that the vanishing theorem follows by \textit{J. Rosenberg} [Publ. Math., Inst. Hautes Étud. Sci. 58, 409--424 (1983; Zbl 0526.53044)].
scientific article; zbMATH DE number 6509751
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