Suppose $S$ is a connected two dimensional manifold(here we assume a manifold to be of countable basis),$s_{0}$ a base point of $S$.
I was asked to show that the following map is an isomorphism:
$$H^{1}(S)\to Hom(\pi_{1}(S,s_{0}),\Bbb{R}),[\alpha]\mapsto \int\alpha$$
The "injective" part is trivial,but how to prove the "surjective" part?
Furthermore, for higher dimensional case,will this ismorphism still be valid?
Any hints or references would be appreciated!