TITLE: Homeomorphism exchanging two homotopic paths
QUESTION [5 upvotes]: Let $X$ be a compact simply connected space and $\gamma_1, \gamma_2 : [0,1] \to X$ be two (homotopic) simple paths between two different points $x,y \in X$. Does there exist a homeomorphism $\varphi : X \to X$ such that $\varphi \circ \gamma_1= \gamma_2$?

REPLY [3 votes]: A positive answer is given by the isotopy extension theorem for smooth manifolds without boundary: if two smooth embeddings of an interval are isotopic, then there is a diffeomorphism of $X$ that interchanges them. If the dimension of $M$ is $\geq 4$, then two embeddings of an interval are isotopic iff they are homotopic.