Dirac index classes and the noncommutative spectral flow
The evaluation on \([M]\) of Hirzebruch's \(L\) polynomial in the Pontryagin classes of a smooth oriented closed manifold \(M\) equals the topological signature \(\sigma(M)\), and so it defines an oriented-homotopy invariant. When \(M\) is not simply connected, ``higher'' signatures can be defined by multiplying \(L\) with the pull-back of a class from \(B\pi_1(M)\). The Novikov conjecture asserts that these higher signatures are homotopy invariants. In the paper under review, Leichtnam and Piazza consider more generally maps \(f:M\to B\Gamma\), where \(B\Gamma\) is the classifying space of a discrete group \(\Gamma\), and Dirac operators on \(M\) twisted with the induced flat bundle \({\mathcal V}_f\), whose fibers are modeled on the reduced \(C^*\)-algebra \(\Lambda\) of the group \(\Gamma\). These Dirac operators are a particular case of the construction of \textit{A.~S.~Mishchenko} and \textit{A.~T.~Fomenko} [Math. USSR, Izv. 15, 87-112 (1980); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 43, 831-859 (1979; Zbl 0416.46052)]. The authors extend to the higher setting the notion of spectral section of \textit{R.~B.~Melrose} and \textit{P.~Piazza} [J. Differ. Geom. 46, No.1, 99-180 (1997; Zbl 0955.58020)], acknowledging earlier unpublished results of F. Wu. This result forms the technical core of the paper. It is needed in the definition of the higher Atiyah-Patodi-Singer- (APS-) and \(b\)-index classes for a manifold with boundary. These two classes are shown to coincide; as a corollary, the difference of the APS-index classes for two spectral projections \({\mathcal P}\) and \({\mathcal P'}\) is shown to equal \([{\mathcal P}-{\mathcal P'}]\in K_0(\Lambda)\). The authors also extend to the higher setting the notion of spectral flow, and give a splitting formula for the index. The final results concern the generalization of the cut-and-paste defect formula for the index of \textit{B.~Booss-Bavnek} and \textit{K.~Wojchiechowski} [C. R. Math. Acad. Sci., Soc. R. Can. 7, 97-102 (1985; Zbl 0575.58032)]. As an application, the authors reprove the cut-and-paste invariance of the Novikov higher signatures using the same assumptions as \textit{E.~Leichtnam, W.~Lück} and \textit{M.~Kreck} [Topology 41, 725-744 (2002; Zbl 0997.57040)].
scientific article; zbMATH DE number 1944979
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