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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeAug 15th 2016
• (edited Aug 15th 2016)

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

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeAug 16th 2016
• (edited Aug 16th 2016)

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 $\mathcal{L}X/S^1$?

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeAug 16th 2016

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.

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeAug 16th 2016

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.

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeOct 15th 2019

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)

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeOct 15th 2019

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

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeJul 2nd 2021
• (edited Jul 2nd 2021)

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.)

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeJul 2nd 2021

• 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)
• CommentRowNumber9.
• CommentAuthorUrs
• CommentTimeJul 2nd 2021

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)
• CommentRowNumber10.
• CommentAuthorUrs
• CommentTimeAug 20th 2021

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.

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