Two topological spaces $X,Y$ have the same type of Homotopy if there exists functions $f:X\to Y$ and $g:Y\to X$ such that $f\circ g = id$ and $g\circ f = id$. In this case, $f, g$ are called Homotopy equivalences, and we denote $X\approx_f Y$.
Show that if $f:X\to Y$ is an Homotopy equivalence, then $f$ induces an isomorphism $f_{*}:H_n(X)\to H_n(Y)$, for each $n\in\mathbb{N}$.
I'm having some trouble working with this material, can anyone provide me a hint or solution?
On
You need to use (1) the functoriality of homology and (2) the homotopy invariance of singular homology which says that, if $f \simeq g : X \longrightarrow Y$ are homotopic maps, then they induce the same morphisms in all homology groups, $f_* = g_* : H_*(X) \longrightarrow H_*(Y)$. Warning: you don't have to prove these results (functoriality, homotopy invariance), just use them.
Hint: Well $H_n$ is a functor! So apply this functor to your compositions, and see what you can conclude!
To be specific if $f \colon X \to Y$is a homotopy equivalence with homotopy inverse $g \colon Y \to X$, then $gf \simeq 1_X$ and $fg \simeq 1_Y$. So applying the functor we get, $f_*g_*$ is the identity and $g_* f_*$ is also the identity, hence $f_*$ is an isomorphism as claimed.