I'm trying to show that the first singular homology group $H_1(\mathbb{S}^1)$ is non trivial. In order to do so, my first goal is to show that if $\sigma\in\mathrm{Sin}_1(\mathbb{S}^1)$, then there exists $\tau:\Delta^1\rightarrow\mathbb{R}$ such that $\sigma=p\circ\tau$, where $p:\mathbb{R}\rightarrow\mathbb{S}^1$ is the covering map given by $t\mapsto\exp(2\pi i t)$. I can show the existence using the fundamental theorem of covering maps (I believe it is called like that).
Q1: The problem appears when I try to define a morphism $\phi:\mathrm{Sin}_1(\mathbb{S}^1)\rightarrow\mathbb{R}$ that maps $\sigma\mapsto d_1\tau-d_0\tau$, because I don't know how to show that it is well defined. I have tried using the fact that if there exists another $\tau':\sigma=p\circ\tau'$, then $d_i(\sigma)=d_i(p\circ\tau)=d_i(p\circ\tau')$ and therefore there exists $m_i\in\mathbb{Z}$ such that $\tau'\circ\delta^i=\tau\circ\delta^i+m_i$. My goal is to see that $m_1-m_0=0$, but I don't know how. Any ideas?
Q2: Assuming that it is well defined, we can extend it to a homomorphism $\phi:C_1(\mathbb{S}^1)=\mathbb{Z}\mathrm{Sin}_1(\mathbb{S}^1)\rightarrow\mathbb{R}$, and my intention now is to show that $\phi\circ\delta:C_2(\mathbb{S}^1)\rightarrow\mathbb{R}$ is the trivial map, but I haven't been able to get there from the definition. How can I do it?
From that point, it is clear that we can define a map $H_1(\mathbb{S}^1)\rightarrow \mathbb{R}$ by sending $[c]\mapsto\phi(c)$, that it is well defined and since it maps $[\sigma_0:(t_0,t_1)\mapsto\exp(2\pi i t_1 )]\mapsto 1$, the result follows.
Thanks.