Proving two paths are not path homotopic If I wanted to prove that two paths $f:[0,1] \rightarrow X$ and $g:[0,1] \rightarrow X$ are not path homotopic, I can prove that $f * \bar{g}$ is not path homotopic with the constant path that stays at $f(0)$. Here $\bar{g}$ denotes the reverse of $g$.
I'm not sure why showing $f * \bar{g}$ is not nulhomotopic tells us that $f$ and $g$ is not path homotopic. Can someone explain this?
Commenter: That is what the term &quot;path-homotopic&quot; means in every reference.
OP: If f and g are path-homotopic, then f * (g-bar) is based homotopic to g * (g-bar) which is based homotopic to the identity, as you saw when you introduced fundamental groups. Contradiction.
Commenter: That is what the term &quot;path-homotopic&quot; means in every reference.
OP: If f and g are path-homotopic, then f * (g-bar) is based homotopic to g * (g-bar) which is based homotopic to the identity, as you saw when you introduced fundamental groups. Contradiction.