Cellular Homology at the Bottom Dimension

Question

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?

Answer 1

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 $X_1$.