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In Smith's paper on homotopy groups for Lorentz manifolds, he builds the loop space of all timelike loops in the following fashion :

  • Consider all piecewise continuous timelike curves which start and end at point $x$. This include timelike curves with $q$ changes in time orientation (the tangent vector of the end of one segment has an opposite time orientation to the beginning of the next)
  • Also include in the group stings based at $x$, which are made from arbitrary paths $\gamma$ in the following way : a sting is a curve of the form $\gamma \ast \gamma^{-1}$, with $\gamma(0) = x$.
  • Include insertions of stings on paths. For a path $\gamma$, consider a point $y$ in $\gamma$, and decompose it in two paths $\gamma = \gamma_+ \ast \gamma_-$, with $\gamma_+(1) = y$. The insertion of a sting $f \ast f^{-1}$ at $y$ is $\gamma^* = \gamma_+\ast f\ast f^{-1} \ast \gamma_-$.
  • The constant path is also included in it, $e(\lambda) = x$

The loop space is then defined by all those elements, and the path composition $\ast$ has a group structure.

The motivation for the inclusion of stings given seems to be that it permits the group structure (although that's not stated clearly either), but that doesn't seem correct, as the constant path and timelike curves seem like enough for that. What is the purpose of adding stings to the loop space? All curves involved will be equivalent to a stingless curve anyway.

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Note that the author defines the loop space $T_q$ to be the space generated by loops with $q$ corners. You want to show that $ff^{-1}\sim e$ in the timelike sense. But $ff^{-1}$ will have at least $2q$ corners since you get corners from each copy. So you include all of the curves of this form in your definition of $T_q$.

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