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David C
David C
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When $X$ is a Riemannian manifold you can build are 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.

When $X$ is a Riemannian you can build are 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 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.

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.

Source Link
David C
David C
  • 9.9k
  • 3
  • 31
  • 58

When $X$ is a Riemannian you can build are 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 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.