A model of the Borel construction on the free loopspace

Abstract

Let $X$ be a CW-complex with basepoint. We obtain a simple description of the Borel construction on the free loopspace of the suspension of $X$ as a wedge of the classifying space of the circle and the homotopy colimit of a diagram consisting of products of a number of copies of $X$ and the standard topological $n$-simplex. This is obtained by filtering the cyclic bar construction on the James model of the based loopspace by word length in order to express the homotopy type of the free loopspace as a colimit of powers of $X$ and standard cyclic sets. It is shown that this colimit is in fact a homotopy colimit and commutativity of homotopy colimits is used to describe the Borel construction.