# 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 $$X_1$$.
• Is there some easy way to see that their boundary operators are the same too? I'm kind of able to convince myself that it's true by analyzing the definition differential in the l.e.s. for the pair and how singular chains for the pair $(X_1, X_0)$ look like but that's not very neat. Commented May 22 at 11:42