This is related to this MO question, I don't know if it's really "research-level". As in that question, let $U$ be a domain of the complex plane $\mathbb{C}$, i.e. an open connected subset. Let

$\kappa_a:=\frac{1}{(\zeta-a)}d\zeta\;\;,\;\;a\in\mathbb{C}$

be the Cauchy kernel centered at $a\in \mathbb{C}$. Of course if $a \in \mathbb{C}-U$, $\kappa_a$ is a holomorphic 1-form on $U$. Given a continuous closed path $\gamma:[0,1]\to U$, $\gamma(1)=\gamma(0)$, we say that $\gamma$ *is equivalent to* $0$ *in* $U$ if for every holomorphic function $f\in\mathcal{O}(U)$, one has

$\langle f\cdot\kappa_a,\gamma\rangle:=\int_\gamma f\cdot \kappa_a=0$

for every $a\notin U$.

1) Is it true that $\gamma$ is equivalent to $0$ in $U$ if and only if $[\gamma]=0\in H_1(U,\mathbb{Z})$ ?

A simple question on the homology of $U$:

2) Is it true that $H_1(U,\mathbb{Z})$ is the free module generated by $[\gamma_i]$, for $\gamma_i$, $i\in\pi_0(\mathbb{C}-U)$, loops each of which goes around each bounded connected component $S_i$ of the complement of $U$ exactly once, i.e. $\langle\kappa_{a_j},\gamma_i\rangle=\delta_{ij}$ for any choice of $a_j\in S_j$? [Edit: not at all sure it may be true: thinking of when $U$ is the complement of a Cantor set... But perhaps a weakened but still meaningful assertion holds?]

By Poincaré duality for non-compact manifolds, the homology (with complex coefficients) $H_i(U,\mathbb{C})$ is dual to de Rham cohomology with compact supports $H^i_{dR,\; c}(U,\mathbb{C})$. (BTW, the latter, see the other question, is isomorphic to the cohomology of the complex $\Gamma(U,\Omega^{\bullet})$ of holomorphic differential forms).

3) Is it possible to find 1-forms $\alpha_i$ on $U$, $i\in\pi_0(\mathbb{C}-U)$, with compact support, such that $\alpha_i$ is cohomologous to $\kappa_{a_i}$ and the $[\alpha_i]$ generate over $\mathbb{C}$ the cohomology $H^1_{dR,\; c}(U,\mathbb{C})$ ? (I'd say

no) Anyway, how to see question (1) in the light of Poincaré duality and/or the remarks above on cohomology?