Bordism, rho-invariants and the Baum-Connes conjecture

Let G be a finitely generated discrete group. In this paper we establish vanishing results for rho-invariants associated to (i) the spin-Dirac operator of a spin manifold with positive scalar curvature (ii) the signature operator of the disjoint union of a pair of homotopy equivalent oriented manifolds with fundamental group G. The invariants we consider are more precisely - the Atiyah-Patodi-Singer rho-invariant associated to a pair of finite dimensional unitary representations. - the L2-rho invariant of Cheeger-Gromov - the delocalized eta invariant of Lott for a finite conjugacy class of G. We prove that all these rho-invariants vanish if the group G is torsion-free and the Baum-Connes map for the maximal group C^*-algebra is bijective. For the delocalized invariant we only assume the validity of the Baum-Connes conjecture for the reduced C^*-algebra. In particular, the three rho-invariants associated to the signature operator are, for such groups, homotopy invariant. For the APS and the Cheeger-Gromov rho-invariants the latter result had been established by Navin Keswani. Our proof re-establishes this result and also extends it to the delocalized eta-invariant of Lott. Our method also gives some information about the eta-invariant itself (a much more saddle object than the rho-invariant).Comment: LaTeX2e, 60 pages; the gap pointed out by Nigel Higson and John Roe is now closed and all statements of the first version of the paper are proved (with some small refinements

Rho-classes, index theory and Stolz' positive scalar curvature sequence

In this paper, we study the space of metrics of positive scalar curvature using methods from coarse geometry. Given a closed spin manifold M with fundamental group G, Stephan Stolz introduced the positive scalar curvature exact sequence, in analogy to the surgery exact sequence in topology. It calculates a structure group of metrics of positive scalar curvature on M (the object we want to understand) in terms of spin-bordism of BG and a somewhat mysterious group R(G). Higson and Roe introduced a K-theory exact sequence in coarse geometry which contains the Baum-Connes assembly map, with one crucial term K(D*G) canonically associated to G. The K-theory groups in question are the home of interesting index invariants and secondary invariants, in particular the rho-class in K_*(D*G) of a metric of positive scalar curvature on a spin manifold. One of our main results is the construction of a map from the Stolz exact sequence to the Higson-Roe exact sequence (commuting with all arrows), using coarse index theory throughout. Our main tool are two index theorems, which we believe to be of independent interest. The first is an index theorem of Atiyah-Patodi-Singer type. Here, assume that Y is a compact spin manifold with boundary, with a Riemannian metric g which is of positive scalar curvature when restricted to the boundary (and with fundamental group G). Because the Dirac operator on the boundary is invertible, one constructs a delocalized APS-index in K_* (D*G). We then show that this class equals the rho-class of the boundary. The second theorem equates a partitioned manifold rho-class of a positive scalar curvature metric to the rho-class of the partitioning hypersurface.Comment: 39 pages. v2: final version, to appear in Journal of Topology. Added more details and restructured the proofs, correction of a couple of errors. v3: correction after final publication of a (minor) technical glitch in the definition of the rho-invariant on p6. The JTop version is not correcte

The surgery exact sequence, K-theory and the signature operator

The main result of this paper is a new and direct proof of the natural transformation from the surgery exact sequence in topology to the analytic K-theory sequence of Higson and Roe. Our approach makes crucial use of analytic properties and new index theorems for the signature operator on Galois coverings with boundary. These are of independent interest and form the second main theme of the paper. The main technical novelty is the use of large scale index theory for Dirac type operators that are perturbed by lower order operators.Comment: 29 pages, AMS-LaTeX; v2: small corrections and (hopefully) improved exposition, as suggested by the referee. Final version, to appear in Annals of K-Theor

Different amplitude and time distribution of the sound of light and classical music

Several pieces of different musical kinds were studied measuring $N(A)$, the output amplitude of a peak detector driven by the electric signal arriving to the loudspeaker. Fixed a suitable threshold $\bar{A}$, we considered $N(A)$, the number of times that $A(t)>\bar{A}$, each of them we named event and $N(t)$, the distribution of times $t$ between two consecutive events. Some $N(A)$ and $N(t)$ distributions are displayed in the reported logarithmic plots, showing that jazz, pop, rock and other popular rhythms have noise-distribution, while classical pieces of music are characterized by more complex statistics. We pointed out the extraordinary case of the aria ``\textit{La calunnia \`{e} un venticello}'', where the words describe an avalanche or seismic process, calumny, and the rossinian music shows $N(A)$ and $N(t)$ distribution typical of earthquakes.Comment: 3 pages with 4 figures, to be published in The European Physical Journal

Eta cocycles, relative pairings and the Godbillon-Vey index theorem

We prove a Godbillon-Vey index formula for longitudinal Dirac operators on a foliated bundle with boundary; in particular, we define a Godbillon-Vey eta invariant on the boundary-foliation; this is a secondary invariant for longitudinal Dirac operators on type-III foliations. Moreover, employing the Godbillon-Vey index as a pivotal example, we explain a new approach to higher index theory on geometric structures with boundary. This is heavily based on the interplay between the absolute and relative pairings of K-theory and cyclic cohomology for an exact sequence of Banach algebras which in the present context takes the form $0\to J\to A\to B\to 0$, with J dense and holomorphically closed in the C^*-algebra of the foliation and B depending only on boundary data. Of particular importance is the definition of a relative cyclic cocycle $(\tau_{GV}^r,\sigma_{GV})$ for the pair $A\to B$; $\tau_{GV}^r$ is a cyclic cochain on A defined through a regularization, \`a la Melrose, of the usual Godbillon-Vey cyclic cocycle $\tau_{GV}$; $\sigma_{GV}$ is a cyclic cocycle on B, obtained through a suspension procedure involving $\tau_{GV}$ and a specific 1-cyclic cocycle (Roe's 1-cocycle). We call $\sigma_{GV}$ the eta cocycle associated to $\tau_{GV}$. The Atiyah-Patodi-Singer formula is obtained by defining a relative index class \Ind (D,D^\partial)\in K_* (A,B) and establishing the equality =<\Ind (D,D^\partial), [\tau^r_{GV}, \sigma_{GV}]>$. The Godbillon-Vey eta invariant$\eta_{GV}$is obtained through the eta cocycle$\sigma_{GV}$.Comment: 86 pages. This is the complete article corresponding to the announcement "Eta cocycles" by the same authors (arXiv:0907.0173