Sending each basis element $a,b,c,d$ to $y-x$.

Question

Hatcher P99 last paragraph:

Define a homomorphism $\partial: C_1 \to C_0$ by sending each basis element $a,b,c,d$ to $y-x$, the vertex at the head of the edge minus the vertex at the tail.

So I am confused: $x,y$ are just vertices, how can they subtract?

http://www.math.cornell.edu/~hatcher/AT/ATch2.pdf

Answer 1

$C_0$ is defined to be the free abelian group generated by the vertices; that is,

$$C_0 = \{n x + m y : n, m \in \mathbb{Z}\}$$

under the natural definitions of $+$ and $-$ for group elements of this form. So $x - y$ is simply the element corresponding to $n = 1$ and $m = -1$ in this group.