We recall some facts about Rational Homotopy Theory, from both Sullivan and Quillen points of view. We show how to find a Sullivan model of a homogeneous space G/H . Let M be a closed oriented smooth manifold of dimension d and LM= map(S^1, M) denote the space of free loops on M . Using intersection products, Chass and Sullivan defined a product on \\mathbb{H}**(LM)=H*{*+d} (LM) that turns to be a graded commutative algebra and defined a bracket on \\mathbb{H}_*(LM) making of it a Gerstenhaber algebra. From the work of Jones, Cohen, F\\â€˜elix, Thomas and others there is an isomorphism of Gerstenhaber algebras between the Hochschild cohomology HH^*(C^*(M), C^*(M)) and \\mathbb{H}_*(LM) . Using a Sullivan model (\\land V, d) of M , we show that that the Gerstenhaber bracket can be computed in terms of derivations on (\\land V, d) . Precisely, we show that HH^*(\\land V, \\land V) is isomorphic to H_*(\\land(V)\\otimes \\land Z , D) , where Z is the dual of V . We will illustrate with computations for homogeneous spaces.

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- Seminars for 2014

Last updated: 29 Nov 2023