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David Chataur

Publications and source records attributed to David Chataur.

25 records · Page 2Linked to original sources
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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arXiv2003

A bordism approach to string topology

Using intersection theory in the context of Hilbert manifolds and geometric homology we show how to recover the main operations of string topology built by M. Chas and D. Sullivan. We also study and build an action of the homology of reduced Sullivan's chord diagrams on the singular homology of free loop spaces, extending previous results of R. Cohen and V. Godin and unifying part of the rich algebraic structure of string topology as an algebra over the prop of these reduced diagrams. Some of these operations are extended to spaces of maps from a sphere to a compact manifold.

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arXiv2003

Fibrewise nullification and the cube theorem

Our aim is to construct fibrewise localizations in model categories. For pointed spaces, the general idea is to decompose the total space of a fibration as a diagram over the category of simplices of the base and replace it by the localized diagram. This of course is not possible in an arbitrary category. We have thus to adapt another construction which heavily depends on Mather's cube theorem. Working with model categories in which the cube theorem holds, we characterize completely those who admit a fibrewise nullification. As an application we get fibrewise plus-construction and fibrewise Postnikov sections for algebras over an operad.

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arXiv2003

Plus-construction for algebras over an operad, cyclic and Hochschild homologies up to homotopy

In this paper we apply homotopical localization to the framework of differential graded algebras over an operad. We get plus construction by performing nullification with respect to an universal acyclic algebra. This plus construction for Lie and Leibniz algebras up to homotopy yields two new homology theories for associative algebras. They are closely related to cyclic and Hochschild homologies. We also compute their first homology groups.

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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

Formes differentielles generalisees sur une operade et modeles algebriques des fibrations

On construit des foncteurs de formes differentielles generalisees. Ceux-ci, dans le cas d'espaces nilpotents de type fini, determinent le type d'homotopie faible des espaces. Ils sont munis, d'une maniere elementaire et naturelle, de l'action de cup-i produits. Pour les algebres commutatives a homotopit pres (algebres sur une resolution cofibrante de l'operade des algebres commutatives), on demontre en utilisant les formes differentielles generalisees que le modele de la fibre d'une application simpliciale est la cofibre du modele de ce morphisme. 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.

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