10. Definition 2.9.4 - chains of closed curves

Let $γ_{1}, \dots, γ_{m}$ be curves, and let $k_{1}, \dots, k_{m}$ be integers. A formal sum:

γ = k_{1} γ_{1} + \dots + k_{m} γ_{m}

is called a $1 - c h a i n$ . By definition, we have:

\oint_{g} f d z = k_{1} \oint_{γ_{1}} f d z + \dots + k_{m} \oint_{γ_{m}} f d z

We can define addition in a natural way (component wise). If every curve $γ_{i}$ is actually closed, then we say $γ$ is a closed $1$ -chain. We define the winding number of a chain in the natural way. For some point $a \in C - ⋃ i m (γ_{i})$ , we have:

n_{γ} (a) = k_{1} n_{γ_{1}} (a) + \dots + k_{m} n_{γ_{m}} (a)

We say that two 1 cycles $γ, η$ are homologous in $U$ , writing $γ \sim η$ , if
:?:

n_{γ} (a) = n_{η} (a) \forall a \in C - U

Furthermore, $γ \sim η$ if $n_{γ} (a) = 0$ for any $a \in C - U$ .