For context, the question popped up while studying the homotopical version of Cauchy's Integral Theorem, and comparing it with the homological version.
Definition: two $γ_0, γ_1 : [0, 1] → U$ be two closed rectifiable curves in a region $G$ are said to be homotopic iff there is a continuous function $Γ : [0, 1] × [0, 1] → G$ such that $$\Gamma(t,0) = \gamma_0(t), \ \ \ \ \Gamma(t,1) = \gamma_1(t), \ \ \ \ \Gamma(0,s) = \Gamma(1,s)$$ for $s,t\in[0,1]$. We say a curve $\gamma$ is homotopic to $0$ iff it is homotopic to the zero curve $z\mapsto 0$.
Definition: two $γ_0, γ_1 : [0, 1] → U$ be two closed rectifiable curves in a region $G$ are said to be homologous iff $w(\gamma_0,z) = w(\gamma_1,z)$ for every $z\notin G$. We say a curve $\gamma$ is homologous to $0$ iff $w(\gamma,z) = 0$ for every $z\notin G$.
1st Question: does being homotopic to $0$ imply being homologous to $0$ or viceversa?
2nd Question: if the first question is answered in the affirmative, then certainly that one of the version of Cauchy's Integral Theorem implies the other is trivial, right?