Define a map as follows: $$ \varphi:H_1(C, \mathbb{Z}) \longrightarrow H^0(\omega_C)^*, \, \, [\gamma]\longmapsto (\omega \longmapsto \int_{\gamma}\omega). $$ where $H^0(\omega_C)$ is the $g$-dimensional $\mathbb{C}$-vector space of holomorphic 1-forms and $H_1(C, \mathbb{Z})$ is the homology group of $C$.
It is a fact that the $\varphi$ map is injective. How to conclude this fact that $H_1(C, \mathbb{Z})$ is a lattice in $H^0(\omega_C)^*$?
Another question is: the quotient $\frac{H^0(\omega_C)^*}{H_1(C, \mathbb{Z})}$ a quotient between groups?
Thanks