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For which $n$ has the $S^1$-equivariant rational cohomology of $LBSO(n)$ been computed? Here, $SO(n)$ means the isometry group of the round sphere (preserving the orientation), $B$ stands for classifying space and $L$ for free loop space. The $S^1$-action on $LBSO(n)$ is, of course, via rotating the loops.

There are two fibrations $$SO(n) \to LBSO(n) \to BSO(n)$$ and $$LBSO(n) \to LBSO(n) \times_{S^1} ES^1 \to BS^1$$ (both admitting a split on the right), and I am interested in knowing $$H^{\ast}_{S^1}(LBSO(n);\mathbb Q) = H^{\ast}(LBSO(n) \times_{S^1} ES^1;\mathbb Q),$$ including the ring structure. So one way to calculate this is to understand what happens in the Serre spectral sequences in the two fibrations above. I could do this (with some effort) for $n = 2,3$ but I do not have much hope figuring out all the differentials for bigger $n$. Is this something that can be found in the literature?

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1st method) If $X$ is a 1-connected then you can use the fact that you have an isomorphism of graded algebras (J.D.S. Jones "Cyclic homology and equivariant homology" Inventiones Math. 1987):

$$H^n_{S^1}(LX;\mathbb{Q})\cong HC_{-n}^{-}(S^*(X;\mathbb{Q}))$$

where the right term is the negative cyclic homology of the cochain algebra of $X$.

Moreover if $X$ is formal you have an isomorphism of algebras:

$$H^n_{S^1}(LX;\mathbb{Q})\cong HC_{-n}^-(H^*(X;\mathbb{Q})).$$

This is indeed the case for $X=BSO(n)$ which is formal. Now you use the fact $H^*(BSO(n;\mathbb{Q}))$ is a polynomial algebra.

2nd method) Rationnaly $BSO(n)$ is formal and homotopy equivalent to a product of Eilenberg-MacLane spaces of type $K(\mathbb{Q},4k)$ and of a copy of $K(\mathbb{Q},2m)$ if $n=2m$, and now use the fact the for an $H$-space $Y$ we have $LY\cong Y\times \Omega Y$. Thus rationnaly you get that the cohomology of $LBSO(n)$ is isomorphic to $$H^*(BSO(n);\mathbb{Q})\otimes H^*(SO(n);\mathbb{Q}).$$

Edit, Reference: A good reference that will give you a cochain complex that computes $H^n_{S^1}(LX;\mathbb{Q})$ is the paper "A model for cyclic homology and algebraic K -theory of 1-connected topological spaces" by Micheline Vigué-Poirrier and Dan Burghelea (J. Differential Geom. Volume 22, Number 2 (1985), 243-253). In particular the theorem $A$ that gives an algebraic model from the minimal model of $X$.

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  • $\begingroup$ Just to add to your 2nd method, to calculate the $S^1$ equivariant $Q$ cohomology of $LBSO(n)$ from that of $LBSO(n)$, you can use that the total space of the fibration $LBSO(n) \to LBSO(n)\times_{S^1} ES^1 \to CP^\infty$ has the following model: Add a generator $u$ in deg 2 (corresponding to $CP^\infty$) to the model of $LBSO(n)$, and extend the differential $d$ from $LBSO(n)$ to this total space (now call it $d'$) by setting $d' u = 0$ and $d' x = dx + u\, i(x)$, where $i$ is the differential on $LBSO(n)$ shifting a generator $x$ of $BSO(n)$ to its corresponding generator in $\Omega BSO(n)$. $\endgroup$
    – Aleksandar Milivojević
    Commented Nov 12, 2018 at 15:18
  • $\begingroup$ This is in Section 9.1. of Chas-Sullivan String topology. The model of the free loop space is explained in projecteuclid.org/euclid.jdg/1214433729 Sullivan-- Vigué-Poirrier. $\endgroup$
    – Aleksandar Milivojević
    Commented Nov 12, 2018 at 15:19

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