Homotopy equivalences

Question

Let $X$ and $Y$ be topological spaces and let $f:X\longrightarrow Y$ and $g:Y\longrightarrow X$ be two continous functions. I should prove that if $f\circ g$ and $g\circ f$ are homotopy equivalences then $f$ and $g$ are homotopy equivalences.

Answer 1

We have that there exists $\alpha,\beta$ such that

$\alpha(fg)\simeq I,$ $(fg)\alpha\simeq I$, $(gf)\beta \simeq I$, $\beta(gf)\simeq I$. Using the second, we have $$ f(g \alpha)\simeq I.$$ Using the last, $$(\beta g)f\simeq I.$$ Now, note that $$ (\beta g)f \simeq I \implies (\beta g) f(g \alpha)\simeq (g \alpha) \implies \beta g \simeq g \alpha.$$

Therefore, $f$ is a homotopy equivalence with inverse given by $\beta g$. Analogously we can prove that $g$ is a homotopy equivalence.