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Rational homotopy of Serre fibrations

Jean-Claude Thomas — 1981

Annales de l'institut Fourier

In rational homotopy theory, we show how the homotopy notion of pure fibration arises in a natural way. It can be proved that some fibrations, with homogeneous spaces as fibre are pure fibrations. Consequences of these results on the operation of a Lie group and the existence of Serre fibrations are given. We also examine various measures of rational triviality for a fibration and compare them with and whithout the hypothesis of pure fibration.

Rational BV-algebra in string topology

Yves FélixJean-Claude Thomas — 2008

Bulletin de la Société Mathématique de France

Let M be a 1-connected closed manifold of dimension m and L M be the space of free loops on M . M.Chas and D.Sullivan defined a structure of BV-algebra on the singular homology of L M , H * ( L M ; 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 H H * ( C * ( M ) ; C * ( M ) ) which extends the canonical structure of Gerstenhaber algebra. We construct then an isomorphism of BV-algebras between H H * ( C * ( M ) ; C * ( M ) ) and the shifted homology H * + m ( L M ; k ) . We also prove that the...

Nilpotent subgroups of the group of fibre homotopy equivalences.

Yves FélixJean-Claude Thomas — 1995

Publicacions Matemàtiques

Let ξ = (E, p, B, F) be a Hurewicz fibration. In this paper we study the space L(ξ) consisting of fibre homotopy self equivalences of ξ inducing by restriction to the fibre a self homotopy equivalence of F belonging to the group G. We give in particular conditions implying that π(L(ξ)) is finitely generated or that L(ξ) has the same rational homotopy type as aut(F).

The Hochschild cohomology of a closed manifold

Yves FelixJean-Claude ThomasMicheline Vigué-Poirrier — 2004

Publications Mathématiques de l'IHÉS

Let M be a closed orientable manifold of dimension and 𝒞 * ( M ) be the usual cochain algebra on M with coefficients in a field. The Hochschild cohomology of M, H H * ( 𝒞 * ( M ) ; 𝒞 * ( M ) ) is a graded commutative and associative algebra. The augmentation map ε : 𝒞 * ( M ) 𝑘 induces a morphism of algebras I : H H * ( 𝒞 * ( M ) ; 𝒞 * ( M ) ) H H * ( 𝒞 * ( M ) ; 𝑘 ) . In this paper we produce a chain model for the morphism I. We show that the kernel of I is a nilpotent ideal and that the image of I is contained in the center of H H * ( 𝒞 * ( M ) ; 𝑘 ) , which is in general quite small. The algebra H H * ( 𝒞 * ( M ) ; 𝒞 * ( M ) ) is expected to be isomorphic...

Rational string topology

Yves FélixJean-Claude ThomasMicheline Vigué-Poirrier — 2007

Journal of the European Mathematical Society

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 simply connected closed manifold M . We prove that the loop homology of M is isomorphic to the Hochschild cohomology of the cochain algebra C * ( M ) with coefficients in C * ( M ) . Some explicit computations of the loop product and the string bracket are given.

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