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We present a detailed proof of the existence-theorem for noncommutative spectral sections (see the noncommutative spectral flow, unpublished preprint, 1997). We apply this result to various index-theoretic situations, extending to the noncommutative context results of Booss– Wojciechowski, Melrose–Piazza and Dai–Zhang. In particular, we prove a variational… (More)

- E. Leichtnam, P. Piazza
- 1998

In this paper we consider Γ → M → M , a Galois covering with boundary and D /, a Γ-invariant generalized Dirac operator on M. We assume that the group Γ is of polynomial growth with respect to a word metric. By employing the notion of noncommutative spectral section associated to the boundary operator D / 0 and the b-calculus on Galois coverings with… (More)

- Eric Leichtnam, Wolfgang LuK, Matthias Kreck
- 2002

We extend the notion of the symmetric signature (M M , r)3¸L(R) for a compact n-dimensional manifold M without boundary, a reference map r : MPBG and a homomorphism of rings with involutions : 9GPR to the case with boundary *M, where (M M , *M)P(M, *M) is the G-covering associated to r. We need the assumption that C H (*M) 9 % R is R-chain homotopy… (More)

- ERIC LEICHTNAM
- 2008

Let Γ be a discrete finitely generated group. Let M → T be a Γ-equivariant fibration, with fibers diffeomorphic to a fixed even dimensional manifold with boundary Z. We assume that Γ → M → M /Γ is a Galois covering of a compact manifold with boundary. Let (D + (θ)) θ∈T be a Γ-equivariant family of Dirac-type operators. Under the assumption that the boundary… (More)

Let X be a compact manifold with boundary ∂X, and suppose that ∂X is the total space of a fibration Z → ∂X → Y. Let DΦ be a generalized Dirac operator associated to a Φ-metric gΦ on X. Under the assumption that DΦ is fully elliptic we prove an index formula for DΦ. The proof is in two steps: first, using results of Melrose and Rochon, we show that the index… (More)

Let N be a closed connected spin manifold admitting one metric of positive scalar curvature. In this paper we use the higher eta-invariant associated to the Dirac operator on N in order to distinguish metrics of positive scalar curvature on N. In particular, we give sufficient conditions, involving π 1 (N) and dim N, for N to admit an infinite number of… (More)

Let (N, g) be a closed Riemannian manifold of dimension 2m − 1 and let → N → N be a Galois covering of N. We assume that is of polynomial growth with respect to a word metric and that N is L 2-invertible in degree m. By employing spectral sections with a symmetry property with respect to the-Hodge operator, we define the higher eta invariant associated with… (More)

- ERIC LEICHTNAM
- 1994

Let X and Y be two closed connected Riemannian manifolds of the same dimension and φ : S * X → S * Y a contact diffeomorphism. We show that the index of an elliptic Fourier operator Φ associated with φ is given by B * (X) e θ0ˆA(T * X) − B * (Y) e θ0ˆA(T * Y) where θ 0 is a certain characteristic class depending on the principal symbol of Φ and B * (X) and… (More)

We discuss the behaviour of the signature index class of closed foliated bundles under the operation of cutting and pasting. Along the way we establish several index theoretic results: we define Atiyah-Patodi-Singer (≡ APS) index classes for Dirac-type operators on foliated bundles with boundary; we prove a relative index theorem for the difference of two… (More)

Building on the theory of elliptic operators, we give a unified treatment of the following topics: • the problem of homotopy invariance of Novikov's higher signatures on closed manifolds; • the problem of cut-and-paste invariance of Novikov's higher signatures on closed manifolds; • the problem of defining higher signatures on manifolds with boundary and… (More)