Poincaré duality and commutative differential graded algebras

Let M be a simply connected closed manifold and consider the (ordered) configuration space F.M; k/ of k points in M . In this paper we construct a commutative differential graded algebra which is a potential candidate for a model of the rational homotopy type of F.M; k/. We prove that our model it is at least a † k -equivariant differential graded model.We also study Lefschetz duality at the level of cochains and describe equivariant models of the complement of a union of polyhedra in a closed manifold. 55P62, 55R80

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“…The main result of [12] states that any closed oriented simply connected manifold admits a CDGA-model which is a connected Poincaré duality CDGA.…”

Section: Notationmentioning

confidence: 99%

“…In [12] we have proved that any simply connected manifold M admits a CDGA model, .A; d /, such that A is a Poincaré duality algebra of dimension m D dim M . We can then define a diagonal class…”

Section: Introductionmentioning

confidence: 99%

A remarkable DGmodule model for configuration spaces

Lambrechts

Stanley

2008

Algebr. Geom. Topol.

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“…Now by a recent result of Lambrechts and Stanley [20] there is a commutative differential graded algebra A satisfying: We call A a Poincaré duality model for M .…”

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confidence: 99%

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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“…Over the rationals it is possible to lift this Frobenius algebra structure to the level of cochains. By a result of Lambrechts and Stanley [LS08], over rationals there is a connected finite dimensional commutative DG algebra A which is quasi-isomorphic to the singular cochain algebra C * (M, Q) on a given n-dimensional manifold M , equipped with a bimodule isomorphism A → A ∨ inducing the Poincaré duality Proof. 1) It follows from the characterization (2.5).…”

Section: Definition 21 (Dg Open Frobenius Algebra)mentioning

confidence: 99%

“…By Félix-Thomas [Fél] theorem, this cup product on HH * (A, A ∨ ) provides an algebraic model for the Chas-Sullivan product on H * (LM ) the homology of the free loop space of closed oriented manifold M . Here one must work over a field of characteristic zero and for A one can take the closed (commutative) Frobenius algebra provided by Lambreschts-Stanley result [LS08] on the existence of an algebraic model with Poincaré duality for the rational singular cochain algebra of a closed oriented maniflold.…”

Section: Co-leibniz Identitymentioning

confidence: 99%

On the Hochschild homology of open Frobenius algebras

Abbaspour¹

2016

J. Noncommut. Geom.

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Abstract. We prove that the shifted Hochschild chain complex C * (A, A) [m] of a symmetric open Frobenius algebra A of degree m has a natural homotopy coBV-algebra structure. As a consequence HH * (A, A) [m] and HH * (A, A ∨ )[−m] are respectively coBV and BV algebras. The underlying coalgebra and algebra structure may not be resp. counital and unital. We also introduce a natural homotopy BV-algebra structure on C * ( is an open Frobenius algebras. If A is commutative, we also introduce a natural BV structure on the shifted relative Hochschild homology HH * (A)[m− 1]. We conjecture that the product of this BV structure is identical to the GoreskyHingston[GH09a] product on the cohomology of free loop spaces when A is a commutative cochain algebra model for M .

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Poincaré duality and commutative differential graded algebras

Cited by 55 publications

References 13 publications

A remarkable DGmodule model for configuration spaces

A remarkable DGmodule model for configuration spaces

Rational BV-algebra in string topology

On the Hochschild homology of open Frobenius algebras

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