Rational BV-algebra in String Topology

Let $M$ be a 1-connected closed manifold and $LM$ be the space of free loops on $M$. In \cite{C-S} M. Chas and D. Sullivan defined a structure of BV-algebra on the singular homology of $LM$, H_\ast(LM; \bk). When the field of coefficients is of characteristic zero, we prove that there exists a BV-algebra structure on \hH^\ast(C^\ast (M); C^\ast (M)) which carries the canonical structure of Gerstenhaber algebra. We construct then an isomorphism of BV-algebras between \hH^\ast (C^\ast (M); C^\ast (M)) and the shifted H_{\ast+m} (LM; {\bk}). We also prove that the Chas-Sullivan product and the BV-operator behave well with the Hodge decomposition of $H_\ast (LM)$

On the cohomology algebra of free loop spaces

Let $X$ be a simply connected space and $\Bbb K$ be any field. The normalized singular cochains $N^*(X; {\Bbb K})$ admit a natural strongly homotopy commutative algebra structure, which induces a natural product on the Hochschild homology $HH_* N^*X$ of the space $X$. We prove that, endowed with this product, $HH_*N^*X$ is isomorphic to the cohomology algebra of the free loop space of $X$ with coefficients in $\Bbb K$. We also show how to construct a simpler Hochschild complex which allows direct computation.Comment: 21 pages, to appear in Topolog

String topology on Gorenstein spaces

The purpose of this paper is to describe a general and simple setting for defining $(g,p+q)$-string operations on a Poincar\'e duality space and more generally on a Gorenstein space. Gorenstein spaces include Poincar\'e duality spaces as well as classifying spaces or homotopy quotients of connected Lie groups. Our presentation implies directly the homotopy invariance of each $(g,p+q)$-string operation as well as it leads to explicit computations.Comment: 30 pages and 2 figure

Real-Time Nearfield Acoustic Holography: Implementation of the Direct and Inverse Impulse Responses in the Time-Wavenumber Domain

The aim of the study is to demonstrate that some methods are more relevant for implementing the Real-Time Nearfield Acoustic Holography than others. First by focusing on the forward propagation problem, different approaches are compared to build the impulse response to be used. One of them in particular is computed by an inverse Fourier transform applied to the theoretical transfer function for propagation in the frequency-wavenumber domain. Others are obtained by directly sampling an analytical impulse response in the time-wavenumber domain or by additional low-pass filtering. To estimate the performance of each impulse response, a simulation test involving several monopoles excited by non stationary signals is presented and some features are proposed to assess the accuracy of the temporal signals resulting from reconstruction processing on a forward plane. Then several inverse impulse responses used to solve the inverse problem, which consists in back propagating the acoustic signals acquired by the microphone array, are built directly from a transfer function or by using Wiener inverse filtering from the direct impulse responses obtained for the direct problem. Another simulation test is performed to compare the signals reconstructed on the source plane. The same indicators as for the propagation study are used to highlight the differences between the methods tested for solving the Holography inverse problem.Comment: 15 th International Congress on Sound and Vibration, Daejeon : Cor\'ee, R\'epublique de (2008