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Let be a 1-connected closed manifold of dimension and be the space of free loops on . M.Chas and D.Sullivan defined a structure of BV-algebra on the singular homology of , . When the ring of coefficients is a field of characteristic zero, we prove that there exists a BV-algebra structure on the Hochschild cohomology which extends the canonical structure of Gerstenhaber algebra. We construct then an isomorphism of BV-algebras between and the shifted homology . We also prove that the...
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 .
We prove that the loop homology of is isomorphic to the Hochschild cohomology of the cochain algebra with coefficients in . Some explicit computations of the loop product and
the string bracket are given.
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...
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 -charges, introduced by De Pauw, Moonens, and Pfeffer: on any subset satisfying a mild geometric condition, there is no uniformly continuous representation operator for -charges in .
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