Is the localised $S^1$-equivariant cohomology of the free loop space of a space $X$ isomorphic to that of $X$ itself?

Question

A well-known theorem of Atiyah and Bott states that given a finite dimensional oriented manifold $M$ with circle action, the $S^1$-equivariant cohomology of $M$ (with $\mathbb{Q}$ coefficients) is isomorphic to the $S^1$-equivariant cohomology of the fixed points after inverting the Euler class of the normal bundle $\nu$ of $i: Fix(M)\hookrightarrow M$ (and shifting degree by the rank of $\nu$). In particular, the isomorphisms are the equivariant pushforward $i_\ast$ in one direction and $i^\ast/eul(\nu)$ in the other. Notice that the $S^1$-equivariant cohomology of a space is a $\mathbb{Q}[u]$-module and $eul(\nu)$ is a multiple of a power of $u$. This means that we just need to localize at $u$ to obtain the theorem.

Now, if we let $X$ be a smooth manifold and $LX$ its free loop space, $S^1$ acts trivially on the former and by loop rotation on the latter. The inclusion map $i:X\hookrightarrow LX$ sending $X$ to the trivial loops is exactly the inclusion of the fixed points. We cannot apply the theorem above for dimensional reasons.

Question 1 Is it still true that ${{H_{S^1}}(LX)}_{(u)} \cong{H_{S^1}(X)}_{(u)}$?

The same could be rephrased as an isomorphism between the localized cyclic homology (see Jones) of $\Omega^\ast(X)$, namely $H\hat{C}_{-\ast}(\Omega^\ast (X))$ and $H^*(X)\otimes \mathbb{Q}[u,u^{-1}]$.

On one hand it seems natural enough for this isomorphism to hold, on the other hand, I have not seen anything stating any connection between them, not even in the paper of Jones in which he defines localized cyclic homology.

Question 2 Is the corresponding statement for localized cyclic homology true? That is, given a cochain complex $S$, is it true that $H\hat{C}_{-*}(S)\cong H^n(S)\otimes\mathbb{Q}[u,u^{-1}]$?

Atiyah, Michael F.; Bott, Raoul, The moment map and equivariant cohomology, Topology 23, 1-28 (1984). ZBL0521.58025.

Jones, John D. S., Cyclic homology and equivariant homology, Invent. Math. 87, 403-423 (1987). ZBL0644.55005.

Answer 1

I don't know what you would mean by the normal bundle of $M$ in $LM$, but the answer to one question is "no": When the $\mathbb Q[u]$-module $H^\ast_{S^1}(LM)$ is localized by inverting $u$, it does not become the same as the cohomology of $M$.

In fact, in my paper in Topology (1985) "Cyclic homology, derivations, and the free loopspace" I prove a very different and incompatible general statement: this localized cohomology depends only on the fundamental group of $M$. More precisely, for a $2$-connected map $X\to Y$ the relative equivariant cohomology of $LX\to LY$ is a torsion $\mathbb Q[u]$-module.

There are related statements involving cyclic homology of differential graded algebras.

Answer 2

As the example by Tom Goodwillie shows, the restriction to fixed points does not generally induce an isomorphism between localized equivariant cohomologies. Yet, as the question shows, there's good reason for willing this isomorphism to exist despite the evidence it does not. The question has been solved by Jones and Petrack in The fixed point theorem in equivariant cohomology. The following extract from their Introduction beautifully explains what they do:

"Now let $X$ be a smooth, connected, finite dimensional manifold and let $LX$ be the space of all smooth loops in $X$ with its $C^\infty$ topology. The space $LX$ is an infinite dimensional manifold modelled on a Fr'echet space. The circle acts smoothly on LX and the fixed point set of this action is precisley the manifold X, considered as the space of constant loops. According to Goodwillie [10], $u^{-1}H^\bullet_T(LX)$ depends only on $\pi_1(X)$ so that the fixed point theorem, as it stands, cannot be true for $LX$. This has been an obstacle to progress in the study of differential forms and integration on $LX$. Our solution to this difficulty is to construct a new form of equivariant cohomology, denoted $h^\bullet_T(Y)$, which will be used in place of $u^{-1} H^\bullet_T(Y)$. The groups $h^\bullet_T(Y)$ are defined by a simple and natural modification of one definition of periodic equivariant cohomology. If $Y$ is finite dimensional then $h^\bullet_T(Y)=u^{-1} H^\bullet_T(Y)$ but in a large class of infinite dimensional examples, including the case $Y = LX$ , the inclusion $i: F \to Y$ of the fixed point set induces an isomorphism $h^\bullet_T(Y) \to h^\bullet_T(F)$. This cohomology theory is constructed in §1 and the fixed point theorem is stated precisely in §2. Some alternative theories which also satisfy the fixed point theorem are discussed in §3. In §4 we show how to construct an inverse, at the level of differential forms, for the restriction homomorphism $i^\ast$ and following [3, 6, 4, and 7] explain how this leads to integration formulas. The most important ingredient is the construction of an equivariant form $\tau$, essentially the form $\exp( -( \omega + E))$ of [2, 6], which defines a class in $h^\bullet_T(Y)$ and has the key property that $i^\ast(\tau) = 1$. This form $\tau$ is essentially the equivariant Thom/Euler class of the normal bundle to the fixed point set. In §5 we show how the form $\tau$ on $LX$, is related to the $\hat{A}$ polynomial of $X$, compare [2]."