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I am trying to compute equivariant (co)homology of the free loop space of a manifold $M$ that is not a Lie group, $H^{S^1}_*(LM)$ with the natural rotation action of $S^1$ on the loops of the free loop space $LM$. In particular, I am interested in integer coefficients.

Is there any advantage to using the isomorphic cyclic homology instead? Do we have extra tools in cyclic homology that can help in the computations of the equivariant (co)homology groups?

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up vote 8 down vote accepted

If one has an understanding of $H_*(LM)$ to begin with, then the bar (or Borel, or Rothenberg-Steenrod) spectral sequence

$$Tor^{\Lambda[\Delta]}_{**}(k, H_*(LM)) \implies H_*^{S^1}(LM)$$

allows you to compute $H_*^{S^1}(LM)$; here the exterior algebra $\Lambda[\Delta] = H_*(S^1)$ uses the group structure on $S^1$. In cyclic homology, there is a nearly identical spectral sequence that's often called "Connes' spectral sequence."

One problem that shows up in computing the $E_2$-term of each of these spectral sequences is how the Batalin-Vilkovisky operator $\Delta$ (or Connes' $B$ operator) coming from the fundamental class of the circle actually acts on $H_*(LM)$ (respectively, the Hochschild cohomology of $C^*(M)$). One advantage to cyclic homology is that $B$ has a very concrete description in terms of Hochschild cochains.

If you can find explicit cocycles which serve as ring generators for $HH^*(C^*(M), C^*(M))$, then you can often compute the action of $B$ directly on these classes. You can then lift this to a computation of the action of $B$ (or $\Delta$) to an action on the entirety of $HH^*(C^*(M), C^*(M))$ (respectively the string topology ring $H_*(LM)$) using the Batalin-Vilkovisky structure on either of these.

My main point is that computing $\Delta$ on low-dimensional classes can be sometimes easier to do using Hochschild homology, and sometimes easier using topology. Then the BV structure allows you to extend these computations to the rest of $H_*(LM)$.

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Thanks Craig! I will try this and see if it gets me anywhere. –  Felix T. Apr 30 '14 at 4:17
Good luck! If it's helpful, I did some simple calculations of this sort here: –  Craig Westerland Apr 30 '14 at 14:48

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