arXiv subjects

Luc Menichi

Publications and source records attributed to Luc Menichi.

17 recordsLinked to original sources

arXiv2016

The bv algebra in string topology of classifying spaces

For almost any compact connected Lie group $G$ and any field $\mathbb{F}\_p$, we compute the Batalin-Vilkoviskyalgebra $H^{*+\text{dim }G}(LBG;\mathbb{F}\_p)$ on the loop cohomology of the classifying space introduced byChataur and the second author.In particular, if $p$ is odd or $p=0$, this Batalin-Vilkovisky algebra is isomorphicto the Hochschild cohomology $HH^*(H\_*(G),H\_*(G))$. Over $\mathbb{F}\_2$, such isomorphism of Batalin-Vilkovisky algebrasdoes not hold when $G=SO(3)$ or $G=G\_2$.

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arXiv2013

String Topology, Euler Class and TNCZ free loop fibrations

Let $M$ be a connected, closed oriented manifold. Let $ω\in H^m(M)$ be its orientation class. Let $χ(M)$ be its Euler characteristic. Consider the free loop fibration $ΩM\buildrel{i}\over\hookrightarrow LM\buildrel{ev}\over\twoheadrightarrow M$. For any class $a\in H^*(LM)$ of positive degree, we prove that the cup product $χ(M)a\cup ev^*(ω)$ is null. In particular, if $i^*:H^*(LM;\mathbb{F}_p)\twoheadrightarrow H^*(ΩM;\mathbb{F}_p)$ is onto then $χ(M)$ is divisible by $p$ (or $M$ is a point).

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arXiv2013

Rational homotopy -- Sullivan models

This chapter is a short introduction to Sullivan models. In particular, we find the Sullivan model of a free loop space and use it to prove the Vigué-Poirrier-Sullivan theorem on the Betti numbers of a free loop space.

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arXiv2013

Derived string topology and the Eilenberg-Moore spectral sequence

Let $M$ be any simply-connected Gorenstein space over any field. Félix and Thomas have extended to simply-connected Gorenstein spaces, the loop (co)products of Chas and Sullivan on the homology of the free loop space $H_*(LM)$. We describe these loop (co)products in terms of the torsion and extension functors by developing string topology in appropriate derived categories. As a consequence, we show that the Eilenberg-Moore spectral sequence converging to the loop homology of a Gorenstein space admits a multiplication and a comultiplication with shifted degree which are compatible with the loop product and the loop coproduct of its target, respectively. We also define a generalized cup product on the Hochschild cohomology $HH^*(A,A^\vee)$ of a commutative Gorenstein algebra $A$ and show that over $\mathbb{Q}$, $HH^*(A_{PL}(M),A_{PL}(M)^\vee)$ is isomorphic as algebras to $H_*(LM)$. Thus, when $M$ is a Poincaré duality space, we recover the isomorphism of algebras $\mathbb{H}_*(LM;\mathbb{Q})^\cong HH^*(A_{PL}(M),A_{PL}(M))$ of Félix and Thomas.

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arXiv2013

Behavior of the Eilenberg-Moore spectral sequence in derived string topology

The purpose of this paper is to give applications of the Eilenberg-Moore type spectral sequence converging to the relative loop homology algebra of a Gorenstein space, which is introduced in the previous paper due to the authors. Moreover, it is proved that the spectral sequence is functorial on the category of simply-connected Poincaré duality spaces over a space.

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arXiv2010

A Batalin-Vilkovisky algebra morphism from double loop spaces to free loops

Let $M$ be a compact oriented $d$-dimensional smooth manifold and $X$ a topological space. Chas and Sullivan \cite{Chas-Sullivan:stringtop} have defined a structure of Batalin-Vilkovisky algebra on $\mathbb{H}_*(LM):=H_{*+d}(LM)$. Getzler \cite{Getzler:BVAlg} has defined a structure of Batalin-Vilkovisky algebra on the homology of the pointed double loop space of $X$, $H_*(Ω^2 X)$. Let $G$ be a topological monoid with a homotopy inverse. Suppose that $G$ acts on $M$. We define a structure of Batalin-Vilkovisky algebra on $H_*(Ω^2BG)\otimes\mathbb{H}_*(M)$ extending the Batalin-Vilkovisky algebra of Getzler on $H_*(Ω^2BG)$. We prove that the morphism of graded algebras $$H_*(Ω^2BG)\otimes\mathbb{H}_*(M)\to\mathbb{H}_*(LM)$$ defined by Felix and Thomas \cite{Felix-Thomas:monsefls}, is in fact a morphism of Batalin-Vilkovisky algebras. In particular, if $G=M$ is a connected compact Lie group, we compute the Batalin-Vilkovisky algebra $\mathbb{H}_*(LG;\mathbb{Q})$.

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arXiv2010

Connes-Moscovici characteristic map is a Lie algebra morphism

Let $H$ be a Hopf algebra with a modular pair in involution $(\Character,1)$. Let $A$ be a (module) algebra over $H$ equipped with a non-degenerated $\Character$-invariant $1$-trace $τ$. We show that Connes-Moscovici characteristic map $φ_τ:HC^*_{(\Character,1)}(H)\rightarrow HC^*_λ(A)$ is a morphism of graded Lie algebras. We also have a morphism $Φ$ of Batalin-Vilkovisky algebras from the cotorsion product of $H$, $\text{Cotor}_H^*({\Bbbk},{\Bbbk})$, to the Hochschild cohomology of $A$, $HH^*(A,A)$. Let $K$ be both a Hopf algebra and a symmetric Frobenius algebra. Suppose that the square of its antipode is an inner automorphism by a group-like element. Then this morphism of Batalin-Vilkovisky algebras $Φ:\text{Cotor}_{K^\vee}^*(\mathbb{F},\mathbb{F})\cong \text{Ext}_{K}(\mathbb{F},\mathbb{F}) \hookrightarrow HH^*(K,K)$ is injective.

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arXiv2010

Van Den Bergh isomorphisms in String Topology

Let $M$ be a path-connected closed oriented $d$-dimensional smooth manifold and let ${\Bbbk}$ be a principal ideal domain. By Chas and Sullivan, the shifted free loop space homology of $M$, $H_{*+d}(LM)$ is a Batalin-Vilkovisky algebra. Let $G$ be a topological group such that $M$ is a classifying space of $G$. Denote by $S_*(G)$ the (normalized) singular chains on $G$. Suppose that $G$ is discrete or path-connected. We show that there is a Van Den Bergh type isomorphism $$ HH^{-p}(S_*(G),S_*(G))\cong HH_{p+d}(S_*(G),S_*(G)). $$ Therefore, the Gerstenhaber algebra $HH^{*}(S_*(G),S_*(G))$ is a Batalin-Vilkovisky algebra and we have a linear isomorphism $$HH^{*}(S_*(G),S_*(G))\cong H_{*+d}(LM).$$ This linear isomorphism is expected to be an isomorphism of Batalin-Vilkovisky algebras. We also give a new characterization of Batalin-Vilkovisky algebra in term of derived bracket.

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arXiv2009

String topology of classifying spaces

Let $G$ be a finite group or a compact connected Lie group and let $BG$ be its classifying space. Let $\mathcal{L}BG:=map(S^1,BG)$ be the free loop space of $BG$ i.e. the space of continuous maps from the circle $S^1$ to $BG$. The purpose of this paper is to study the singular homology $H_*(\mathcal LBG)$ of this loop space. We prove that when taken with coefficients in a field the homology of $\mathcal LBG$ is a homological conformal field theory. As a byproduct of our main theorem, we get a Batalin-Vilkovisky algebra structure on the cohomology $H^*(\mathcal LBG)$. We also prove an algebraic version of this result by showing that the Hochschild cohomology $HH^*(S_* (G),S_*(G))$ of the singular chains of $G$ is a Batalin-Vilkovisky algebra.

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arXiv2007

Batalin-Vilkovisky algebra structures on Hochschild Cohomology

Let $M$ be any compact simply-connected $d$-dimensional smooth manifold and let $\mathbb{F}$ be any field. We show that the Gerstenhaber algebra structure on the Hochschild cohomology on the singular cochains of $M$, $HH^*(S^*(M);S^*(M))$, extends to a Batalin-Vilkovisky algebra. Such Batalin-Vilkovisky algebra was conjecturated to exist and is expected to be isomorphic to the Batalin-Vilkovisky algebra on the free loop space homology on $M$, $H_{*+d}(LM)$ introduced by Chas and Sullivan. We also show that the negative cyclic cohomology $HC^*_-(S^*(M))$ has a Lie bracket. Such Lie bracket is expected to coincide with the Chas-Sullivan string bracket on the equivariant homology $H_*^{S^1}(LM)$.

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arXiv2007

String topology for spheres

Let $M$ be a compact oriented $d$-dimensional smooth manifold. Chas and Sullivan have defined a structure of Batalin-Vilkovisky algebra on $\mathbb{H}_*(LM)$. Extending work of Cohen, Jones and Yan, we compute this Batalin-Vilkovisky algebra structure when $M$ is a sphere $S^d$, $d\geq 1$. In particular, we show that $\mathbb{H}_*(LS^2;\mathbb{F}_2)$ and the Hochschild cohomology $HH^{*}(H^*(S^2);H^*(S^2))$ are surprisingly not isomorphic as Batalin-Vilkovisky algebras, although we prove that, as expected, the underlying Gerstenhaber algebras are isomorphic. The proof requires the knowledge of the Batalin-Vilkovisky algebra $H_*(Ω^2 S^3;\mathbb{F}_2)$ that we compute in the Appendix.

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arXiv2007

Batalin-Vilkovisky algebras and the J-homomorphism

Let X be a topological space. The homology of the iterated loop space $H_*Ω^n X$ is an algebra over the homology of the framed n-disks operad $H_*f\mathcal{D}_n$ \cite{Getzler:BVAlg,Salvatore-Wahl:FrameddoBVa}. We determine completely this $H_*f\mathcal{D}_n$-algebra structure on $H_*(Ω^n X;\mathbb{Q})$. We show that the action of $H_*(SO(n))$ on the iterated loop space $H_*Ω^n X$ is related to the J-homomorphism and that the BV-operator vanishes on spherical classes only in characteristic other than 2.

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arXiv2003

Batalin-Vilkovisky algebras and cyclic cohomology of Hopf algebras

We show that the Connes-Moscovici cyclic cohomology of a Hopf algebra equipped with a character has a Lie bracket of degree -2. More generally, we show that a "cyclic operad with multiplication" is a cocyclic module whose cohomology is a Batalin-Vilkovisky algebra and whose cyclic cohomology is a graded Lie algebra of degree -2. This explain why the Hochschild cohomology algebra of a symmetric algebra is a Batalin-Vilkovisky algebra.

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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

On the cohomology algebra of a fiber

Let f:E-->B be a fibration of fiber F. Eilenberg and Moore have proved that there is a natural isomorphism of vector spaces between H^*(F;F_p) and Tor^{C^*(B)}(C^*(E),F_p). Generalizing the rational case proved by Sullivan, Anick [Hopf algebras up to homotopy, J. Amer. Math. Soc. 2 (1989) 417--453] proved that if X is a finite r-connected CW-complex of dimension < rp+1 then the algebra of singular cochains C^*(X;F_p) can be replaced by a commutative differential graded algebra A(X) with the same cohomology. Therefore if we suppose that f:E-->B is an inclusion of finite r-connected CW-complexes of dimension < rp+1, we obtain an isomorphism of vector spaces between the algebra H^*(F;F_p) and Tor^{A(B)}(A(E),F_p) which has also a natural structure of algebra. Extending the rational case proved by Grivel-Thomas-Halperin [PP Grivel, Formes differentielles et suites spectrales, Ann. Inst. Fourier 29 (1979) 17--37] and [S Halperin, Lectures on minimal models, Soc. Math. France 9-10 (1983)] we prove that this isomorphism is in fact an isomorphism of algebras. In particular, $H^*(F;F_p) is a divided powers algebra and p-th powers vanish in the reduced cohomology \mathaccent "707E {H}^*(F;F_p).

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arXiv2001

P-th powers in mod p cohomology of fibers

Let $F\hookrightarrow E\twoheadrightarrow B$ be a fibration whose base space $B$ is a finite simply-connected CW-complex of dimension $\leq p$ and whose total space $E$ is a path-connected CW-complex of dimension $\leq p-1$. If $α\in H^{+}(F;\mathbb{F}_p)$ then $α^{p}=0$.

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arXiv2000

The cohomology ring of free loop spaces

Let X be a simply connected space and k a commutative ring. Goodwillie, Burghelea and Fiedorowiscz proved that the Hochschild cohomology of the singular chains on the pointed loop space HH^{*}S_*(ΩX) is isomorphic to the free loop space cohomology H^{*}(X^{S^{1}}). We proved that this isomorphism is compatible with both the cup product on HH^{*}S_*(ΩX) and on H^{*}(X^{S^{1}}). In particular, we explicit the algebra H^{*}(X^{S^{1}}) when X is a suspended space, a complex projective space or a finite CW-complex of dimension p such that \frac {1}{(p-1)!}\in k.

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