It is well-known that a based space is of the weak homotopy type of a loop space iff it is a grouplike algebra over an $A_{\infty }$-operad. The classical model for such an operad consists of Stasheff’s associahedra. The present paper describes a similar recognition principle for free loop spaces. Let $𝒫$ be an operad, $M$ a $𝒫$-module and $U$ a $𝒫$-algebra. An $M$-trace over $U$ consists of a space $V$ and a module homomorphism $T:M\to {\text{End}}_{U,V}$ over the operad homomorphism $𝒫\to {\text{End}}_U$ given by the algebra structure on $U$. Let ${𝒞}_1$ be the little 1-cubes operad.The author shows that the free loop space $\wedge X$ is a trace over the ${𝒞}_1$-space $\Omega X$. This trace is related to the cyclohedra in a way similar to the relation of ${𝒞}_1$ to the associahedra. Given a $𝒫$-module $M$ and a $𝒫$-algebra $U$ one can define the free $M$-trace over $U$ like one can construct free $𝒫$-al!

EUDML-ID : urn:eudml:doc:220881
Mots clés:

@article{701714,
     title = {Free loop spaces and cyclohedra},
     booktitle = {Proceedings of the 22nd Winter School "Geometry and Physics"},
     series = {GDML\_Books},
     publisher = {Circolo Matematico di Palermo},
     address = {Palermo},
     year = {2003},
     pages = {151-157},
     mrnumber = {MR1982442},
     zbl = {1032.55006},
     url = {http://dml.mathdoc.fr/item/701714}
}

Markl, Martin. Free loop spaces and cyclohedra, dans Proceedings of the 22nd Winter School "Geometry and Physics", GDML_Books,  (2003), pp. 151-157. http://gdmltest.u-ga.fr/item/701714/