A higher Atiyah-Patodi-Singer index theorem for the signature operator on Galois coverings
The authors [Geom. Funct. Anal. 8, No. 1, 17-58 (1998; Zbl 0913.58055)] have shown that in general higher index theory associated with Galois coverings of manifolds with boundary depends on the choice of a boundary condition known as a spectral section. The paper under review considers the extent to which a signature operator's higher index is independent of the choice of spectral section. Assume that the base manifold's fundamental group has polynomial growth, and assume that the boundary's cover has a Laplacian that is \(L^{2}\)-invertible in the middle degrees. In this setting, the authors describe a class of spectral sections they call symmetric, and they show that the signature operator can always be assigned a symmetric spectral section. Furthermore the signature operator's higher index is independent of the choice of symmetric spectral section. The authors give a higher Atiyah-Patodi-Singer index formula by expressing the Chern character of the signature operator's higher index class as an explicitly defined element in the noncommutative de Rham homology of a suitable completion of the fundamental group's group algebra. One term in this index formula is known as a higher eta invariant. The authors' proof of their index formula is based on the \(b\)-calculus of R. Melrose. Spectral sections were introduced by \textit{R. B. Melrose} and \textit{P. Piazza} [J. Differ. Geom. 46, No. 1, 99-180 (1997) and J. Differ. Geom. 46, No. 2, 287-334 (1997; Zbl 0920.58053)] and were first defined in a noncommutative setting in a preprint of F. Wu. Much of the framework for higher Atiyah-Patodi-Singer index theory is due to \textit{J. Lott} [Geom. Funct. Anal. 2, No. 4, 421-454 (1992; Zbl 0777.58038) and K-Theory 6, No. 3, 191-233 (1992; Zbl 0773.58026)].
scientific article; zbMATH DE number 1436919