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Summary:
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 ${\cal P}$ be an operad, $M$ a ${\cal P}$-module and $U$ a ${\cal P}$-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 ${\cal P}\to\text{End}_U$ given by the algebra structure on $U$. Let ${\cal C}_1$ be the little 1-cubes operad.\par The author shows that the free loop space $\wedge X$ is a trace over the ${\cal C}_1$-space $\Omega X$. This trace is related to the cyclohedra in a way similar to the relation of ${\cal C}_1$ to the associahedra. Given a ${\cal P}$-module $M$ and a ${\cal P}$-algebra $U$ one can define the free $M$-trace over $U$ like one can construct free ${\cal P}$-al!
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