If $X$ is a based connected topological space, it is well-known what the homology of $\Omega\Sigma X$ is: according to the Bott-Samelson theorem, it is a tensor algebra over reduced homology of $X$. (Some assumptions about coefficients should be put here).
However, it is also known that $\Omega\Sigma X$ is weakly equivalent to $D_1X$, where $D_1$ is a monad associated to little intervals operad. Passing to the homology, I can see that:
- $H_*(\mathcal{D}_1(*);R)$ forms an operad in graded $R$-modules;
- Homology of $D_1X$ is an algebra over this operad.
All of the proofs of the Bott-Samelson theorem which I know use rather geometric description of $\Omega\Sigma X$, without using any information coming from the operad action. So my (probably quite vague) question is: is it possible to prove B-S theorem using operadic data? The result being something like "free $\mathcal{D}_1$ algebra in $R$-modules with base $H_*(X;R)$"?
Maybe this is a question about reference, but any help would be appreciated.
