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Rational BV-algebra in string topology

Yves Félix, Jean-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...

Rational string topology

Yves Félix, Jean-Claude Thomas, Micheline 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.

Relationship among various Vietoris-type and microsimplicial homology theories

Takuma Imamura (2021)

Archivum Mathematicum

In this paper, we clarify the relationship among the Vietoris-type homology theories and the microsimplicial homology theories, where the latter are nonstandard homology theories defined by M.C. McCord (for topological spaces), T. Korppi (for completely regular topological spaces) and the author (for uniform spaces). We show that McCord’s and our homology are isomorphic for all compact uniform spaces and that Korppi’s and our homology are isomorphic for all fine uniform spaces. Our homology shares...

Retractions onto the Space of Continuous Divergence-free Vector Fields

Philippe Bouafia (2011)

Annales de la faculté des sciences de Toulouse Mathématiques

We prove that there does not exist a uniformly continuous retraction from the space of continuous vector fields onto the subspace of vector fields whose divergence vanishes in the distributional sense. We then generalise this result using the concept of m -charges, introduced by De Pauw, Moonens, and Pfeffer: on any subset X ⊆ ℝ n satisfying a mild geometric condition, there is no uniformly continuous representation operator for m -charges in X .

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