Homeomorphism exchanging two homotopic paths 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 $\varphi \circ \gamma_1= \gamma_2$?
 A: Let $X$ be a closed disk, let $x$ and $y$ be two points on its boundary, let $\gamma_1$ be a path from $x$ to $y$ along the boundary, and let $\gamma_2$ be a path from $x$ to $y$ through the interior of the disk.  A homeomorphism from the disk onto itself takes the boundary to the boundary, so it can't send $\gamma_1$ to $\gamma_2$.
A: I don't think so.  Let $X = S^2\vee S^3$, the one point union of $S^2$ and $S^3$.  Since both $S^2$ and $S^3$ are compact and simply connected, so is $X$.  Let $\gamma_1$ be a path in $S^2$ (starting at the wedge point, if you want) and $\gamma_2$ a path in $S^3$.
I claim there is no homeomorphism $\phi$ moving $\gamma_1$ to $\gamma_2$.  The point is that any homeomorphism of $X$ must send the wedge point to itself, and then this implies the $S^2$ must be sent to itself and likewise the $S^3$ must be sent to itself.
A: 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.
