Is loop spaces are homotopy equivalent

Question

Let X be a path connected space and suppose a and b are two points of X. If we consider the loop space at “a” say (A,a) and loop space at “b” say (B,b) and f be a path from a to b. Then is it true that (A,a) and (B,b) are homotopy equivalent? Certainly there is canonical map from one to other using f.

Answer 1

Yes this is true. From a loop $\gamma$ based at $a$ and your $f$ you can define the loop $f\# \gamma\# \overline f$. Here $\overline f$ denotes the inverse path. This map $\Psi_f:\Omega_a(X)\rightarrow \Omega_b(X)$ is continuous in the compact open topology. The inverse of the map is given by $\Psi_\overline f$. The loops $f\#\overline f$ and $\overline f\#f$ are contractible, and it is not too hard to use these contractions to show that $\Psi_f\circ \Psi_\overline f$ and $\Psi_\overline f\circ \Psi_f$ are homotopic to the identity on $\Omega_a(X)$ and $\Omega_b(X)$ respectively.