Differential Graded Algebras in Topology

Félix, Yves; Halperin, Stephen; Thomas, Jean‐Claude

doi:10.1016/b978-044481779-2/50017-1

Cited by 100 publications

(92 citation statements)

References 24 publications

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“…is an R-free module on a basis of cycles (see [7]). Then, If P is a differential graded right A e -module then the cochain complex…”

Section: Hochschild Homology and Cohomologymentioning

confidence: 99%

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Rational BV-algebra in string topology

Félix¹,

Thomas²

2008

Bul. Soc. Math. France

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Abstract. -Let M be a 1-connected closed manifold of dimension m and LM be the space of free loops on M . M. Chas and D. Sullivan defined a structure of BValgebra on the singular homology of LM , H * (LM ; k). When the ring of coefficients is a field of characteristic zero, we prove that there exists a BV-algebra structure on the Hochschild cohomology HH * (C * (M ); C * (M )) which extends the canonical structure of Gerstenhaber algebra. We construct then an isomorphism of BV-algebras between HH * (C * (M ); C * (M )) and the shifted homology H * +m (LM ; k). We also prove that the Chas-Sullivan product and the BV-operator behave well with a Hodge decomposition of H * (LM ).Texte reçu le 7 juin

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“…is an R-free module on a basis of cycles (see [7]). Then, If P is a differential graded right A e -module then the cochain complex…”

Section: Hochschild Homology and Cohomologymentioning

confidence: 99%

“…We use rational homotopy theory for which we refer systematically to [7]. We only recall here that a morphism in some category of complexes is a quasi-isomorphism if it induces an isomorphism in homology.…”

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confidence: 99%

Rational BV-algebra in string topology

Félix¹,

Thomas²

2008

Bul. Soc. Math. France

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“…On peut généraliser la notion d'extension libre (on parle aussi de morphisme quasi-libre) introduite dans le cadre des algèbres différentielles graduées [12] aux O-algèbres.…”

Section: Extensions Libres Dans Cette Section On Caractérise Les Cofiunclassified

Formes différentielles généralisées sur une opérade et modèles algébriques des fibrations

Chataur

2002

Algebr. Geom. Topol.

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We construct functors of generalized differential forms. In the case of nilpotent spaces of finite type, they determine the weak homotopy type of the spaces. Moreover they are equipped, in an elementary and natural way, with the action of cup-i products. Working with commutative algebras up to homotopy (viewed as algebras over a cofibrant resolution of the operad of commutative algebras), we show using these functors that the model of the fiber of a simplicial map is the cofiber of the algebraic model of this map.Resumé On construit des foncteurs de formes différentielles généralisées. Ceux-ci, dans le cas d'espaces nilpotents de type fini, déterminent le type d'homotopie faible des espaces. Ils sont munis, d'une manièreélémentaire et naturelle, de l'action de cup-i produits. Pour les algèbres commutatives a homotopit prés (algèbres sur une résolution cofibrante de l'opérade des algèbres commutatives), on démontre en utilisant les formes différentielles généralisées que le modèle de la fibre d'une application simpliciale est la cofibre du modèle de ce morphisme.

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“…By [10], Proposition 6.13 in the case F D pt, when G is a topological group or by [10], Theorem 6.3, when G D M , there exists a differential graded coalgebra B.S .EG/I S .G/I k/ and two quasi-isomorphisms of coalgebras…”

Section: Is a Quasiisomorphism If And Only If The Morphism Of Rightmentioning

confidence: 99%