Spectral Sections and Higher Atiyah-Patodi-Singer Index Theory on Galois Coverings

In this paper we consider $ \Gamma \to \tilde M \to M $ a Galois covering with boundary and $ \not \tilde D $ , $ \Gamma $ -invariant generalized Dirac operator on $ \tilde M $ . We assume that the group $ \Gamma $ is of polynomial growth with respect to a word metric. By employing the notion of noncommutative spectral section associated to the boundary operator $ \not \tilde D_0 $ and the b-calculus on Galois coverings with boundary, we develop a higher Atiyah-Patodi-Singer index theory. Our main theorem extends to such $ \Gamma $ -Galois coverings with boundary the higher index theorem of Connes-Moscovici.