arXiv subjects

Jean-Claude Thomas

Publications and source records attributed to Jean-Claude Thomas.

12 recordsLinked to original sources

arXiv2007

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) $.

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arXiv2005

Exponential growth of Lie algebras of finite global dimension

Let $X$ be a finite simply connected CW complex of dimension $n$. The loop space homology $H\_*(ΩX;\mathbb Q)$ is the universal enveloping algebra of a graded Lie algebra $L\_X$ isomorphic with $ pi\_{*-1} (X)\otimes \mathbb Q$. Let $Q\_X \subset L\_X$ be a minimal generating subspace, and set $α= \limsup\_i \frac{\log{\scriptsize rk} π\_i(X)}{i}$. Theorem: If ${dim} L\_X = \infty$ and $\limsup ({dim} (Q\_X)\_k)^{1/k} < \limsup ({dim} (L\_X)\_k)^{1/k}$ then $$\sum\_{i=1}^{n-1} {rk} π\_{k+i}(X) = e^{(α+ ε\_k)k} \hspace{1cm} {where} ε\_k \to 0 {as} k\to \infty.$$ In particular $\displaystyle\sum\_{i=1}^{n-1} {rk} π\_{k+i}(X)$ grows exponentially in $k$.

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arXiv2005

Frobenius Rational Loop Algebra

Recently R. Cohen and V. Godin have proved that the homology of the free loop space of a closed oriented manifold with coefficients in a field has the structure of a Frobenius algebra without counit. In this short note we prove that when the characteristic of the field is zero and when the manifold is 1-connected the algebraic structure depends only on the rational homotopy type of the manifold. We build an algebraic model and use it to do some computations.

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arXiv2004

Rational String Topology

We use the computational power of rational homotopy theory to provide an explicit cochain model for the loop product and the string bracket of a 1-connected closed manifold M. We prove that the loop homology of M is isomorphic to the Hochschild cohomology of the commutative graded algebra A_{PL}(M) with coefficients in itself. Some explicit computations of the loop product and the string bracket are given.

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arXiv2003

Loop homology algebra of a closed manifold

The loop homology of a closed orientable manifold $M$ of dimension $d$ is the ordinary homology of the free loop space $M^{S^1}$ with degrees shifted by $d$, i.e. $\mathbb H_*(M^{S^1}) = H_{*+d}(M^{S^1})$. Chas and Sullivan have defined a loop product on $\mathbb H_*(M^{S^1})$ and an intersection morphism $I : \mathbb H_*(M^{S^1}) \to H_*(ΩM)$. The algebra $\mathbb H_*(M^{S^1})$ is commutative and $I$ is a morphism of algebras. In this paper we produce a model that computes the algebra $\mathbb H_*(M^{S^1})$ and the morphism $I$. We show that the kernel of $I$ is nilpotent and that the image is contained in the center of $H_*(ΩM)$, which is in general quite small.

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arXiv2003

Massey products and configuration spaces

The purpose of this paper is to compare two spectral sequences converging to the cohomology of a configuration space. The collapsing of these spectral sequences is established, in some cases, using Massey products.

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arXiv2002

Operadic Hochschild chain complex and free loop spaces

We construct for any algebra over an operad an Hochschild chain complex. In the case of the singular cochain complex of a topological space, considered as a commutative algebra up to homotopy, we show that this complex computes the singular cohomology of the free loop space over this topological space.

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arXiv2002

Duality in Gerstenhaber algebras

Let $C$ be a differential graded coalgebra, $ \barΩC$ the Adams cobar construction and $C^\vee$ the dual algebra. We prove that for a large class of coalgebras $C$ there is a natural isomorphism of Gerstenhaber algebras between the Hochschild cohomologies $HH^\ast (C^\vee, C ^\vee)$ and $HH^\ast (\barΩC ; \barΩC)$. This result permits to describe a Hodge decomposition of the loop space homology of a closed oriented manifold, in the sense of Chas-Sullivan, when the field of coefficients is of characteristic zero.

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arXiv2002

Spaces of self-equivalences and free loops spaces

Let M be a simply-connected closed oriented N-dimensional manifold. We prove that for any field of coefficients there exists a natural homomorphism of commutative graded algebras $Ψ: H_\ast (Ω{aut}_1 M) \to H_{\ast +N}(M^{S^1})$ where $H_\ast (M^{S^1})$ is the loop algebra defined by Chas-Sullivan. As usual ${aut}_1 X$ (resp. $ΩX$) denotes the monoid of the self-equivalences homotopic to the identity map (resp. the space of based loops) of the space X. Moreover, if $\bk$ is of characteristic zero, $Ψ$ yields isomorphisms $π_n(Ω{aut}_1 M) \otimes \bk \cong \hH^{n+N}_{(1)}$ where $\displaystyle \oplus_{l=1}^\infty \hH^n_{(l)}$ denotes the Hodge decomposition on $H^\ast (M ^{S^1})$.

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arXiv2001

Cochain algebras of mapping spaces and finite group actions

The purpose of the present article is threefold. First of all, we rebuild the whole theory of cosimplicial models of mapping spaces by using systematically Kan adjunction techniques. Secondly, given two topological spaces X and Y, we construct a cochain algebra which is quasi-isomorphic (as an algebra) to the singular cochain algebra of the corresponding mapping space from X to Y. Here, X has to be homotopy equivalent to the geometric realization a finite simplicial set and of dimension less or equal to the connectivity of Y. At last, we apply these results to the study of finite group actions on mapping spaces that are induced by an action on the source.

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arXiv2001

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.

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arXiv2001

Steenrod operations and Hochshild homology

Let $X$ be a simply connected space and ${\Bbb F}_p$ be a prime field. The algebra of normalized singular cochains $N^*(X; {\Bbb F}_p)$ admits a natural homotopy structure which induces natural Steenrod operations on the Hochschild homology $HH_* N^*(X;{\Bbb F}_p)$ of the space $X$. The primary purpose of this paper is to prove that the J. Jones isomorphism $HH_*N^*(X;{\Bbb F}_p) \cong H ^*(X^{S^1};{\Bbb F}_p)$ identifies theses Stenrood operations with those defined on the cohomology of the free loop space with coefficients in ${\Bbb F}_p$. The other goal of this paper is to describe a theoritic model which allows to do some computations.

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