Higher eta-invariants
The eta-invariant was introduced by \textit{M. F. Atiyah}, \textit{V. K. Patodi} and \textit{I. M. Singer} [Math. Proc. Camb. Philos. Soc. 77, 43-69 (1975; Zbl 0297.58008)] to prove an index theorem for elliptic differential operators on manifolds with boundary. The present paper deals with the definition of a higher eta-invariant for Dirac type operators, this is one which does involve the fundamental group \(\Gamma\) of the manifold. In the first part of the paper it is assumed that \(\Gamma\) contains an Abelian subgroup of finite index. Using an approach of \textit{G. Lusztig} for the study of Novikov's higher signature [J. Differ. Geom. 7, 229-256 (1972; Zbl 0265.57009)] a certain fibration is associated to the manifold, and the eta form \(\widetilde\eta\) is defined as a differential form on the base of the fibration. In this setting a higher index theorem for manifolds with boundary is stated, based on results of \textit{J.-M. Bismut} and \textit{J. Cheeger} [J. Funct. Anal. 89, No. 2, 313-363 (1990; Zbl 0696.53021) and 90, No. 2, 306-354 (1990; Zbl 0711.53023)]. An application is given to the question whether a closed manifold with positive scalar curvature can be the boundary of a manifold with the same property and a product metric near the boundary. The second part of the paper deals with the higher eta-invariant \(\widetilde\eta\) for more general \(\Gamma\). In this case the author uses methods of noncommutative geometry to define \(\widetilde\eta\) as a differential form, such that \(\widetilde\eta\) can be paired with the cyclic cohomology of the group algebra \(\mathbb{C}\Gamma\). Variational properties of \(\widetilde\eta\) are discussed. A conjecture on the higher index theorem for Dirac type operators on a manifold with boundary is stated, and a Novikov conjecture for manifolds with boundary is proposed.
scientific article; zbMATH DE number 149771
scientific article