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.

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