Free loop spaces and cyclohedra. (English) Zbl 1032.55006

Markl, Martin

Bureš, Jarolím (ed.), The proceedings of the 22nd winter school “Geometry and physics”, Srní, Czech Republic, January 12-19, 2002. Palermo: Circolo Matemàtico di Palermo, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 71, 151-157 (2003).

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 \({\mathcal P}\) be an operad, \(M\) a \({\mathcal P}\)-module and \(U\) a \({\mathcal P}\)-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 \({\mathcal P}\to\text{End}_U\) given by the algebra structure on \(U\). Let \({\mathcal C}_1\) be the little 1-cubes operad.
The author shows that the free loop space \(\wedge X\) is a trace over the \({\mathcal C}_1\)-space \(\Omega X\). This trace is related to the cyclohedra in a way similar to the relation of \({\mathcal C}_1\) to the associahedra. Given a \({\mathcal P}\)-module \(M\) and a \({\mathcal P}\)-algebra \(U\) one can define the free \(M\)-trace over \(U\) like one can construct free \({\mathcal P}\)-algebras. Using the previous result and a theorem of R. Cohen the author proves an approximation theorem relating the free \(C\)-trace over \(KX\) to \(\wedge\Sigma X\) where \(K\) is the operad consisting of the associahedra, \(KX\) the free \(K\)-algebra on \(X\), and \(C\) is the \(K\)-module consisting of the cyclohedra.
For the entire collection see [Zbl 1014.00011].

Reviewer: Rainer Vogt (Osnabrück)

MSC:

55P48	Loop space machines and operads in algebraic topology
18D50	Operads (MSC2010)

Keywords:

modules and traces for operad structures; approximation of free loop space; associahedra; cyclohedra