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I want to find some algebraic models relating string topology to Hochschild and cyclic homologies. If the space $X$ is simply-connected and we are working over rational numbers, we can use Sullivan model or Quillen model of $X$ to do the job. See nLab page or this paper.

My question is how about non-simply conncted spaces? We still have Sullivan model. But how to relate them to free loop spaces? Can we still use Hochschild and cyclic homologies to compute loop homology and string homology?

For example, the torus $T^{2}$. Its Sullivan model is just $\wedge (x,\,y)$ with zero differential where $|x|=|y|=1$, and it is isomorphic to the Chevalley-Eilenberg cochain algebra of 2 dimensional abelian Lie algebra. But I don't know how to use this to compute loop and string homologies.

Any help would be very appreciated.

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    $\begingroup$ For the special case of the torus $T^n$, the fibration $\Omega T^n \to LT^n \to T^n$ is trivial: the fiber is contractible to $\Bbb Z^n$, so $LT^n$ is homotopy equivalent to a covering space, and each element of the fiber belongs to a distinct component of $LT^n$ (this uses that $\Bbb Z^n$ is an abelian group, so every conjugacy class is one element), so the covering space is trivial. So $LT^n \simeq \Bbb Z^n \times T^n$. I imagine the $\text{Diff}(S^1)$-equivariant homotopy type is more complicated. $\endgroup$
    – mme
    Commented Aug 3, 2018 at 22:33

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In my paper with Zeinalian (https://arxiv.org/abs/1612.04801) we prove that the coHochschild complex of the dg coalgebra of singular chains on a path connected (possibly non-simply connected) space calculates the homology of the free loop space of $X$.

In my paper with Saneblidze (https://arxiv.org/abs/1712.02644) we describe a smaller model which might be suitable for your calculations.

However, you must be careful: if you use dg algebra models - such as the Sullivan model - under quasi-isomorphisms you loose the information the fundamental group.

If you want to calculate something for a non-simply connected space $X$ you may use any connected dg coalgebra $C$ which is weakly equivalent to the dg coalgebra of singular chains on $X$ with Alexander-Whitney coproduct in the following sense: we say a map $f: C \to C'$ between two dg connected coalgebras is a weak equivalence (we also called $\Omega$-quasi-isomorphism) if the map of dg algebras $\Omega f: \Omega C \to \Omega C'$ obtained after applying the cobar functor is a quasi-isomorphism. Note that this notion is stronger than quasi-isomorphism of dg coalgebras.

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  • $\begingroup$ Thanks for your answer. Maybe a stupid question, for the case of $T^{2}$, does the Chevalley-Eilenberg chain coalgebra of 2 dimensional abelian Lie algebra give you the coalgebra model you mentioned? $\endgroup$
    – Yining Zhang
    Commented Aug 3, 2018 at 20:26
  • $\begingroup$ Is there an obstruction to analysing the cohomology of the free loop space using similar methods? $\endgroup$
    – user2520938
    Commented Apr 12, 2021 at 14:34
  • $\begingroup$ Also, when $X$ is a finite CW complex, can we take the cellular chain complex instead of the singular chain complex? $\endgroup$
    – user2520938
    Commented Apr 12, 2021 at 14:39
  • $\begingroup$ First you must take a dg coalgebra model for the chains on $X$. Then you must make sure that the dg coalgebra model is weakly equivalent to the singular chains on X in the sense that they can be related by maps of dgcoalgebras that induce quasi-isomorphisms of dg algebras after applying the cobar functor. This can be forced by formally inverting 1-simplices in the cobar construction of your model for the chains. $\endgroup$
    – Manuel Rivera
    Commented Apr 12, 2021 at 21:58

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