# Bijection between homotopy classes

Let $$X$$ be a CW-complex and $$w: Z \to Y$$ a weak homotopy equivalence. Show that $$w_*: [X, Z] \to [X, Y]: [f] \mapsto [w \circ f]$$ is a bijection. Hint: use mapping cylinders.

I am having trouble with this question for both the injectiveness and surjectiveness.

Injective

Suppose $$[w \circ f] = [w \circ g]$$. We can factor $$f$$ as $$X \xrightarrow{i_X} M(f) \xrightarrow{p} Y$$ where $$i_X$$ is a closed inclusion and $$p$$ is a homotopy equivalence (and $$M(f)$$ is the mapping cylinder of $$f$$) and similarly factor $$g$$ as $$X \xrightarrow{j_X} M(g) \xrightarrow{q} Y$$.

I want to use Whitehead on $$w$$. The space $$Z$$ is not a CW-complex, but it is homotopy equivalent to $$M(f)$$ and $$M(g)$$, which are. Similarly $$Y$$ is homotopy equivalent to $$M(w \circ f)$$ and $$M(w \circ g)$$. Composing $$w$$ with the homotopy equivalences gives a map $$M(f) \to M(w \circ g)$$ (we can swap $$f$$ and $$g$$ around), and since homotopy equivalences do not change the $$\pi_n$$, this is a weak homotopy equivalence, hence by Whitehead a homotopy equivalence. I don't know what to do with this.

Surjective

Given a map $$f: X \to Y$$, I can't figure out a way to factor it via $$z$$ as $$w$$ has no (partial) inverse.

Use the factorisation $$Z \overset{j_Z}{\to} M(w) \overset{p}{\to} Y$$ of $$w$$ to show that $$j_Z$$ is a weak homotopy equivalence. This implies that $$(M(w), j_Z(Z))$$ is $$n$$-connected for every $$n$$.
• For surjectivity, any $$f : X \to Y$$ gives $$j_Y \circ f : (X, \varnothing) \to (M(w), j_Z(Z))$$. By the above, and that $$X$$ is a CW-complex, this composition will be homotopic to $$j_Z \circ \tilde{f}$$ for some $$\tilde{f} : X \to Z$$. It is now easy to see that $$[f] = [w \circ \tilde{f}] = w_* [\tilde{f}]$$.
• For injectivity, if $$f, g : X \to Z$$ are such that $$[w \circ f] = [w \circ g]$$, let $$H : X \times [0, 1] \to Y$$ be a homotopy between them. We can slightly alter this to obtain $$H': X \times [0, 1] \to M(w) : (x, t) \mapsto \left\{ \begin{array}{cl} (f(x), 3t) & \text{if } 0 \leq t \leq 1/3 \\ H(x, 3t - 1) & \text{if } 1/3 \leq t \leq 1/3 \\ (g(x), 2 - 3t) & \text{if } 2/3 \leq t \leq 1/3 \\ \end{array} \right.$$ and view $$H'$$ as a map of pairs $$H' : (X \times [0, 1], X \times \{ 0, 1 \}) \to (M(w), j_Z(Z))$$. Similar as to before, $$H'$$ will be homotopic (relative to $$X \times \{ 0, 1 \}$$) to some $$\tilde{H} : X \times [0, 1] \to M(w)$$ with image in $$j_Z(Z)$$. Now it is easy to see this gives a homotopy between $$f$$ and $$g$$.