Free Loop Spaces and Cyclohedra

In this note we introduce an action of the module of cyclohedra on the free loop space. We will then discuss how this action can be used in an appropriate recognition principle in the same manner as the action of Stasheff’s associahedra can be used to recognize based loop spaces. We will also interpret one result of R.L. Cohen as an approximation theorem, in the spirit of Beck and May, for free loop spaces. Recall that, while a (topological) operad is a collection P = {P(n)}n≥0 of topological spaces with structure operations (‘compositions’) ◦i : P(k)× P(l) → P(k + l − 1), 1 ≤ i ≤ k, (1) (see [7, 4]), a (right) module over the operad P is a collection M = {M(n)}n≥0 with structure operations (called again ‘compositions’) ◦i : M(k)× P(l) → M(k + l − 1), 1 ≤ i ≤ k, (2) satisfying appropriate axioms which are in fact ‘linearizations’ of that of an operad, see [4, Definition 1.3]. Example 1. Let U and V be topological spaces. The collection EndU,V := {Hom(U , V )}n≥0 is a natural right module over the endomorphism operad EndU = {Hom(U , U)}n≥0. The structure operations are given in the expected manner as (f ◦i g)(u1, . . . , uk+l−1) := f(u1, . . . , ui−1, g(ui, . . . , ul−1+i), ul+i, . . . , uk+l−1), (3) for f : U → V ∈ EndU,V (k), g : U ×l → U ∈ EndU(l) and u1, . . . , uk+l−1 ∈ U . An algebra over the operad P (also called a P-space) is a topological space U together with Σn-equivariant maps An : P(n) → Hom(U , U), n ≥ 0, which are compatible with the compositions. Thus algebras are ‘representations’ of operads. Representations of modules are traces, introduced in [5, Definition 2.6] as follows: Definition 2. Let M be a P-module as above and U a P-algebra. An M-trace over U is a topological space V together with Σn-equivariant maps Tn : M(n) → Hom(U , V ), n ≥ 0, which are compatible with compositions (2) and the P-algebra structure on U , that is Tk+l−1(m ◦i p) = Tk(m) ◦i Al(p), for m ∈ M(k), p ∈ P(l), 1 ≤ i ≤ k, where ◦i in the right hand side is as in (3). 1991 Mathematics Subject Classification. 55P48, 18D650.