Eric Leichtnam

On Higher Eta-Invariants and Metrics of Positive Scalar Curvature

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 π1(N) and dim N ,f orN to admit an infinite number of metrics of positive scalar curvature that are nonbordant.

Dirac index classes and the noncommutative spectral flow

Abstract 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a detailed proof of the existence-theorem for noncommutative spectral sections (see the noncommutative spectral flow, unpublished preprint, 1997).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 formula, in K ∗ (C r ∗ (Γ)) , for the index classes associated to 1-parameter family of Dirac operators on a Γ-covering with boundary; this formula involves a noncommutative spectral flow for the boundary family. Next, we establish an additivity result, in K ∗ (C r ∗ (Γ)) , for the index class defined by a Dirac-type operator associated to a closed manifold M and a map r:M→BΓ when we assume that M is the union along a hypersurface F of two manifolds with boundary M=M + ∪ F M − . Finally, we prove a defect formula for the signature-index classes of two cut-and-paste equivalent pairs (M1,r1:M1→BΓ) and (M2,r2:M2→BΓ), where M 1 =M + ∪ (F,φ 1 ) M − , M 2 =M + ∪ (F,φ 2 ) M − and φj∈Diff(F). The formula involves the noncommutative spectral flow of a suitable 1-parameter family of twisted signature operators on F. We give applications to the problem of cut-and-paste invariance of Novikovs higher signatures on closed oriented manifolds.

Etale groupoids, eta invariants and index theory

Abstract Let Γ be a discrete finitely generated group. Let → T be a Γ-equivariant fibration, with fibers diffeomorphic to a fixed even dimensional manifold with boundary Z. We assume that Γ → → | 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 family is L 2-invertible, we define an index class in K 0(C 0(T ) ⋊ r Γ ). If, in addition, Γ is of polynomial growth, we define higher indices by pairing the index class with suitable cyclic cocycles. Our main result is then a formula for these higher indices: the structure of the formula is as in the seminal work of Atiyah, Patodi and Singer, with an interior geometric contribution and a boundary contribution in the form of a higher eta invariant associated to the boundary family. Under similar assumptions we extend our theorem to any G -proper manifold, with G an étale groupoid. We employ this generalization in order to establish a higher Atiyah-Patodi-Singer index formula on certain foliations with boundary. Fundamental to our work is a suitable generalization of Melrose b -pseudodifferential calculus as well as the superconnection proof of the index theorem on G -proper manifolds recently given by Gorokhovsky and Lott in [A. Gorokhovsky and J. Lott , Local index theory over étale groupoids, J. reine angew. Math. 560 (2003), 151–198].

A Higher Atiyah–Patodi–Singer Index Theorem for the Signature Operator on Galois Coverings

AbstractLet (N, g) be a closed Riemannianmanifold of dimension 2m − 1 and let Γ → Ñ → N be a Galois covering of N. We assumethat Γ is of polynomial growth with respect to a word metric and that ΔÑ is L2-invertible in degree m. By employing spectral sections with asymmetry property with respect to the ⋆-Hodge operator, we define the higher eta invariant associatedwith the signature operator on Ñ, thus extending previous work of Lott. If π1(M)→

The Novikov conjecture on Cheeger spaces

{\tilde M}

Refined intersection homology on non-Witt spaces

→M is the universal cover of a compact orientable even-dimensionalmanifold with boundary (∂M = N)then, under the above invertibility assumption on Δ∂

Higher eta invariants and the Novikov conjecture on manifolds with boundary

{\tilde M}