Free Loops, Moore Paths and the Borel Construction

Question

My question is about the relationship between the free loop space LX of a space X and the (appropriately defined) Borel construction $PX \times_{\Omega X} \Omega X$ which is a homotopy equivalent fibration. The shortest formulation of my question is:

Is this homotopy equivalence fibre-wise deloopable? (equivalently) Is it a fibre-wise $A_\infty$ equivalence?

So,
The claim is that there is a model for the free loop space LX given by the Borel construction. For this we take based Moore loops in X and Moore paths in X starting at the base point. The Borel construction is then the space of equivalence classes $[p,\alpha]$ where $p$ is a Moore path in X, $\alpha$ is a Moore loop at the base point. The equivalence relation is given by $[p \beta, \alpha] = [p, \beta \alpha \beta^{-1}]$ (where $\beta$ is also a loop at base point).

What is the homotopy equivalence between this and LX? Is this homotopy equivalence fibre wise an $A_\infty$ equivalence? What is the algebraic structure on the fibres of the Borel space?

There is a natural map we can write down from the Borel space at least to the space of free Moore loops in which we map $[p,\alpha]$ to $p\alpha p^{-1}$. However the the image of this map is not even closed under composition in each fibre! (just regard the composition of $p\alpha p^{-1}$ and $p'\alpha p^{'-1}$ where $p$ and $p'$ are different.

a.

Answer 1

When $X$ is a Riemannian manifold you can build very nice models of free loop spaces, based loop spaces and path spaces, you can have a look at "On "small geodesics" and free loop spaces" by Bahri and Cohen available on arxiv. This is based on Milnor's papers on classifying spaces.

Thus what you get is a topological group $GX$ which is $A_{\infty}$-equivalent to $\Omega X$ and a homotopy equivalence between the Borel construction of the adjoint action of $GX$ on itself and the free loop space. A point in $GX$ is an equivalence class of a sequence of points $[x_0,\ldots,x_n]$ where $x_0=x_n$ is the based point, $x_i$ and $x_{i+1}$ are close to each other in a geodesic sense, modulo some cancellation rules. The group structure is concatenation of sequences of points. And the map from $GX$ to $\Omega X$ is by considering composition of geodesics segments. You can also build models of path spaces, free loop spaces in this way. With this gadget you will have maps of fibrations compatible to composition of loops and that are homotopy equivalences.

This construction can be adapted to triangulated spaces, the idea uses the fact that you have a notion of close points and a unique geodesics segment between them.