3 problems

Let $M$ be a simply connected compact oriented smooth manifold, let $LM=\operatorname{map}(S^1,M)$ be its free loop space, and write $\mathbb{H}_*(LM)=H_{*+d}(LM)$ for shifted free…

Let $M$ be simply connected and let ${\Bbbk}$ be a field. Let $LM=map(S^1,M)$ be the free loop space, let $d$ be the dimension parameter of $M$, and let $S^*(M)$ denote the algebra…

Let $G$ be a topological group such that $M$ is a classifying space of $G$, and let $S_*(G)$ denote the normalized singular-chain algebra of $G$. Write $LM=map(S^1,M)$ for the free…