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 $X$ be a simply connected space and let $\mathcal{L}X=\operatorname{Map}(S^1,X)$ be its free loop space. Write $\Omega X$ for the based loop space, and call $X$ rationally hype…

Let $X$ be a simply-connected finite CW-complex, let ${\Bbbk}$ be a field, and let $i:\Omega X\rightarrow LX$ be the inclusion of based loops into free loops. Suppose that … is ont…