John Jones (University of Warwick)
jdsj@maths.warwick.ac.uk
A graded commutative algebra is a Gerstenhaber algebra if it has a graded Lie bracket of degree 1 (or -1) which is a graded derivation of the product. Examples include: (1) The Hochschild cohomology of an associative algebra. (2) The homology of the free loop space of a manifold -- this is due to Chass and Sullivan and is known as the string homology of the manifold. (3) The homology of double loop spaces.
In this talk I will outline a proof of the fact that the string homology of a manifold is isomorphic, as Gerstenhaber algebras, to the Hochschild cohomology of the algebra of singular cochains (or differential forms if we are working over the real numbers) of the manifold.
This result relates examples (1) and (2) above. I will also outline some results/ideas relating all three examples.