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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a detailed proof of the existence-theorem for noncommutative spectral sections (see the noncommutative spectral flow, unpublished preprint, 1997).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)

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)

- 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)

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)