After the effort invested in proving the general theorem on acyclic models, it is time to apply it to topology. First, let us prove:

Theorem 5 Suppose ${f, g: X \rightarrow Y}$ are homotopic. Then the maps ${f_*, g_*: H_*(X) \rightarrow H_*(Y)}$ are equal.

Proof: Suppose ${H: X \times [0,1]}$ is a homotopy with ${H(\cdot, 0) = f, H(\cdot, 1) = g}$. Then the maps ${f,g}$ factor as

$\displaystyle X \rightrightarrows^{i_0}_{i_1} X \times [0,1] \rightarrow^H Y$

so if we show that the inclusions ${i_0,i_1}$ sending ${x \in X}$ to ${(x, 0), (x,1)}$ induce equal maps on homology, we will be done.

Write ${I = [0,1]}$ for simplicity. For each space ${X}$, the maps ${i_0, i_1: X \rightarrow X \times I}$ are natural. More precisely, if ${X \rightarrow X, X \rightarrow X \times I}$ are the two functors ${\mathbf{Top} \rightarrow \mathbf{Top}}$, then ${i_0, i_1}$ are natural transformations between them.