Generators of the first singular homology group of a Riemann surface

Question

Let $X$ be a Riemann surface embedded in $\mathbb C^2$ with coordinates $(z,w)$ and let $\pi_z \colon X \to \mathbb C$ be the projection on first coordinate with property that that for cofinite number of points $z \in \mathbb C$ there exists a neighborhood $U_z$, containing $z$, and there exist neighborhoods of points in $\pi^{-1}(z)$ such that $\pi_z$ is the biholomorphic map from each of these neighborhoods to $U_z$. For a finite number of points $z_0 \in \mathbb C$ we allow to have some points in $\pi^{-1}(z_0)$ such that they have neighborhoods with local coordinates $u$ in which the map is just $z-z_0 = u^k$, $k \geq 2$. Now let $\{\gamma_j\}$ be a collection of small closed curves of the form $u = \varepsilon e^{i\varphi}$, $\varphi \in [0,2\pi)$, $\varepsilon$ is small, around these specific points. Is it true that collection $\{\gamma_j\}$ generates $H_1^s(X)$?

Answer 1

No, because these cycles are contractible, i.e. they are null-homologous. Just let $\epsilon$ shrink to $0$.