Let $X$ and $Y$ be topological spaces. A continuous mapping $f : X \to Y$ is said to be a homotopy equivalence if there exists $g : Y \to X$ continuous such that $g\circ f$ is homotopic to $id_{X}$ and $f\circ g$ is homotopic to $id_{Y}$. In such a case, $g$ is called a homotopy inverse of $f$.
Show that if $g_{1}$ and $g_{2}$ are homotopy inverses of a mapping $f$, then $g_{1}$ and $g_{2}$ are homotopic.
I know that two continuous functions are called homotopic if there is a function $F:X \times I \to Y$ with the properties $H(x,0) = f(x)$ and $H(x,1) = g(x)$. The first parameter is our original function, the second one shows the extent of deformation.
How I can use these informations to show that $g_{1}$ and $g_{2}$ are homotopic?
Thanks in advance
Best Answer
If you know that you can compose homotopies, then $$ g_1\simeq g_1\circ (f\circ g_2)= (g_1\circ f)\circ g_2\simeq g_2 $$ where $g_1$ and $g_2$ are the two homotopy inverses.