The cohomology algebra of certain free loop spaces
Let $X$ be a simply connected space and $LX$ the space of free loops on $X$. We determine the mod $p$ cohomology algebra of $LX$ when the mod p cohomology of $X$ is generated by one element or is an exterior algebra on two generators. We also provide lower bounds on the dimensions of the Hodge decomposition factors of the rational cohomology of $LX$ when the rational cohomology of $X$ is a graded complete intersection algebra. The key to both of these results is the identification of an important subalgebra of the Hochschild homology of a graded complete intersection algebra over a field.