There used to be mentioning of “twisted de Rham cohomology” in a sub-section headline, without this ever being mentioned in the text. Now I have instead removed these words from the section headline, but added an actual remark (here) on this point.

]]>also this one:

- Kentaro Matsuo,
*The Borel cohomology of the loop space of a homogeneous space*, Topology and its Applications**160**12 (2013) 1313-1332 (doi:10.1016/j.topol.2013.05.001)

added pointer to:

- Bitjong Ndombol & M. El Haouari,
*The free loop space equivariant cohomology algebra of some formal spaces*, Mathematische Zeitschrift**266**(2010) 863–875 (doi:10.1007/s00209-009-0602-z)

I have harmonized notation by replacing all “`/^h`

” and “`/ /`

” by “`\sslash`

”

(On the other hand, I guess there was a time when `\sslash`

didn’t render properly on some browsers. I forget what the status is, hope it works now.)

changed title, making it just “loop space” instead of “free loop space”

]]>added as a corollary the Sullivan model for based loop spaces (here) and, as an example, the Sullivan models for the iterated based loop spaces of spheres $\Omega^k S^{n}$ for $k \lt n$ (here)

]]>I have added an Examples-section *The 4-sphere and twisted de Rham cohomology* which spells out the Sullivan model for $\mathcal{L}S^4 // S^1$ and makes an observation of how this relates to a kind of caloron correspondence.

I have found a source for the proof of the Sullivan model for $\mathcal{L}X/S^1$. It is theorem A in Vigué-Burghelea 85. I have added the statement here.

]]>There is a canonical $S^1$-action on the free loop space. For Sullivan models $(\wedge^\bullet (V \oplus s V, d_{\mathcal{L}X} )$ such that $d_X$ is simple enough, then it is easy to guess a Sullivan model for the $S^1$-homotopy quotient. Namely add a generator $\omega_2$ of degre 2 and then modify the differential on the unshifted generators by adding a term proportional to the corresponding shifted generator wedge $\omega_2$.

Is there any published statement about Sullivan models for the homotopy quotients $\mathcal{L}X/S^1$?

]]>I have created *Sullivan model of free loop space* with the formula and pointers to the literature.