##### Venue: Lecture Hall I, Department of Mathematics

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\\‘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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