On the real cohomology of spaces of free loops on manifolds

Let LX be the space of free loops on a simply connected manifoldX. When the real cohomology of X is a tensor product of algebras generated by a single element, we determine the algebra structure of the real cohomology of LX by using the cyclic bar complex of the de Rham complex Ω(X) of X. In consequence, the algebra generators of the real cohomology of LX can be represented by differential forms on LX through Chen’s iterated integral map. Let T be the circle group. The T-equivariant cohomology of LX is also studied in terms of the cyclic homology of Ω(X). Introduction. Let X be a simply connected finite-dimensional manifold whose real cohomology is a tensor product of truncated polynomial algebras and exterior algebras. We call such a commutative algebra a TE-algebra. Let LX be the space of free loops on X, that is, the space of all smooth maps from the circle group T to X. The purpose of this paper is to determine the algebra structures of the real cohomology of LX when the real cohomology ring of X is a TE-algebra, and of the T-equivariant real cohomology of LX when the real cohomology of X is isomorphic to that of a sphere. Moreover, we will represent generators of the real cohomology and of the T-equivariant real cohomology of LX by explicit elements in the Hochschild homology and in the cyclic homology of the de Rham complex of X respectively. Let X be a simply connected space and F(X) the fiber square