f~g iff if $f*\bar{g}$~$\varepsilon_x$

Question

Can you please help me with this question?

let f,g:I $\rightarrow$ X be two paths in X from x to y.

Prove that f~g iff $f*\bar{g}$~$\varepsilon_x$

(where $\bar{g}$=g(t-1))

Thanks in advance

Answer 1

Note that $f\ast \bar{g}$ is a path from $x$ to $x$. If $f\sim g$ then can you show that for any path $h$ from $y$ to $x$ we have $f\ast h\sim g\ast h$? (hint: use the path homotopy that exists between $f$ and $g$). Can you show then that the path $g\ast \bar{g}$ is homotopic to the constant path $\varepsilon_x$?

For the other direction, if you know that $f\ast\bar{g}\sim\varepsilon_x$, then try composing on the right by $g$ and again use what you know about homotopic paths and their reverse.