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Cyclic cocycles, renormalization and eta-invariants. (English) Zbl 0721.58049

The authors discuss in a unified context three of the real-valued invariants that are associated to a self-adjoint pseudo-differential operator (\(\psi\) DO) on a manifold:
a) The relative eta-invariant of Atiyah-Patodi-Singer for a first order, elliptic differential operator coupled to a trivialized flat bundle.
b) The odd analogue of the Breuer index for a self-adjoint, leafwise elliptic \(\psi\) DO along the leaves of a measured foliation, as defined by Connes.
c) The odd analogue of the distributional G-index of a self-adjoint differential operator, transversally elliptic for a smooth G-action, based on the transverse index theory of Atiyah and Singer.
The unification of these three classes of real-valued invariants is a consequence of the authors’ study of a long-standing problem posed by I. M. Singer.
Problem I. Construct the relative eta-invariant as a Breuer index in an appropriate von Neumann algebra.
Section I is an introduction. Section 2 gives the details and sets notation for the basic geometric constructions used throughout the monograph. Section 3 gives full details on the construction of the longitudinal cocycle c. It begins by introducing the class of longitudinal (or leafwise) differential operators, then discusses the construction of DD parametrices with controlled supports for these operators. In Section 4 the main result is established: there are appropriate choices of foliated manifolds and leafwise elliptic data so that the topological formulas for the analytic invariants of a) and b) above coincide. Section 5 introduces the concept of a sharp parametrix for a transverse operator. A key result of Section 6 is that the Weyl asymptotic formula implies that renormalization of the cocycle of \(c^{\#}\) transforms it into the longitudinal cocycle of Section 3. Section 7 reformulates the eta-invariant of D with twisted coefficients as a distribution on the central functions R(G). (D is a geometric operator, operating on sections C(E) of a smooth bundle \(E\to M\) of Clifford modules, M is a compact odd-dimensional Riemannian manifold of dimension m without boundary, G is a connected compact Lie group.) In Section 8 the distributional relative eta-invariant to the values of the renormalized transverse cocycle \(\hat c\) is related.

MSC:

58J40 Pseudodifferential and Fourier integral operators on manifolds
47G30 Pseudodifferential operators
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