# Cellular Homology at the Bottom Dimension

In his book Algebraic Topology, Hatcher constructs cellular homology of a CW-complex $$X$$ as the homology of the following chain complex: $$\cdots \to H_{n+1}(X_{n+1},X_n) \overset{d_{n+1}}{\to} H_n(X_n,X_{n-1}) \overset{d_n}{\to} H_{n-1}(X_{n-1},X_{n-2}) \to \cdots$$ where the maps are induced by boundary maps in the long exact sequences of the pairs $$(X_{i+1},X_i)$$. Hatcher claims that when $$n=1$$, computing the map $$d_1 : H_1(X_1,X_0) \to H_0(X_0)$$ is easy because the map is the same as the simplicial boundary map $$\Delta_1(X) \to \Delta_0(X)$$. I'm having trouble seeing this. How do we even show that we can identify $$H_1(X_1,X_0)$$ with $$\Delta_1(X)$$ and $$H_0(X_0)$$ with $$\Delta_0(X)$$? I know that $$H_1(X_1,X_0) \cong H_1^\Delta(X_1,X_0)$$ and $$H_0(X_0) \cong H_0^\Delta(X_0)$$, but I'm stuck there.

Also the notation is throwing me off - How can we even talk about $$\Delta_1(X)$$ and $$\Delta_0(X)$$ when $$X$$ isn't even a $$\Delta$$-complex?

Although $$X$$ may not be a $$\Delta$$-complex, $$X_1$$ certainly is.
Every CW complex of dimension $$\le 1$$ is a $$\Delta$$-complex.
Since $$X_0$$ is a discrete topological space, it follows that $$H_0(X_0)$$ is the direct sum of one copy of $$\mathbb Z$$ for each point in $$X_0$$, which is exactly how $$\Delta_0(X_0)$$ is defined.
Similarly, by this point Hatcher has proved that $$H_1(X_1,X_0)$$ is the direct sum of one copy of $$\mathbb Z$$ for each 1-cell in $$X_1$$, and a 1-cell in $$X_1$$ is the same as a 1-simplex in $$X_1$$, so this is also exactly how $$\Delta_1(X_1)$$ is defined, namely a direct sum of one copy of $$\mathbb Z$$ for each $$1$$-simplex in $$X_1$$.