Homotopy and chain homotopy:

Chain homotopy is the algebraization of the topological notion of homotopic maps. Let $F:I\times X\longrightarrow Y$ be a homotopy between two continuous functions $f : X \longrightarrow Y$ and $g:X\longrightarrow Y$. We use this map to define a sequence of maps

$$L_n:S_n(X)\longrightarrow S_{n+1}(Y)\eqno(33.6)$$

satisfying the condition

$$\partial \circ L_n + L_{n-1}\circ \partial = f_{\sharp} - g_{\sharp}. \eqno(33.7)$$

Let $u:\Delta_1\longrightarrow I$ be the unique one simplex. For a singular $n$ simplex $\sigma$ in $X$, define

$$L_n(\sigma) = F_{\sharp}(u\times \sigma).$$

Then we compute using (33.1)-(33.2),

$$\partial (L_n(\sigma)) = F_{\sharp}(\partial u\times \sigma) - F_{\sharp}(u\times \partial \sigma)$$

$$= F_{\sharp}(\partial u\times \sigma) - L_{n-1}(\partial \sigma)$$

$$\therefore\; \partial (L_n(\sigma)) + L_{n-1}(\partial \sigma) = F_{\sharp}(\{1\}\times \sigma) - F_{\sharp}(\{0\}\times \sigma)$$

$$\therefore\; \partial (L_n(\sigma)) + L_{n-1}(\partial \sigma) = F(1, \sigma(\cdot)) - F(0, \sigma(\cdot)).$$

So we have the important equation

$$\partial (L_n(\sigma)) + L_{n-1}(\partial \sigma) = g_{\sharp}(\sigma) - f_{\sharp}(\sigma),\quad \sigma\in S_n(X). \eqno(33.8)$$

completing the proof of (33.7). The reader must go back to lemma (32.2) to observe some analogies. After these preparations we are ready to prove the following important result. Unlike theorems (11.2) - (11.5) we do not have to worry here about base points which makes life a lot easier.