[proofplan] We use the Farey tessellation of the extended upper half-plane by ideal triangles with vertices in $\mathbb{P}^1(\mathbb{Q})$. Its oriented edges give relative $1$-chains whose images in the quotient compact modular curve are precisely modular symbols, and its oriented triangles give the two Manin relations. Since every oriented edge is an $SL_2(\mathbb{Z})$-translate of the edge from $0$ to $\infty$, the relative homology is generated by the symbols attached to the finite coset set $\Gamma_0(N)\backslash SL_2(\mathbb{Z})$. The edge-reversal symmetry gives the $S$ relation, while the boundary of the standard ideal triangle gives the $U$ relation. [/proofplan] [step:Represent relative cycles by geodesic edges between cusps] Let $\mathfrak{H}$ denote the complex upper half-plane, and let \begin{align*} \mathfrak{H}^* := \mathfrak{H}\cup \mathbb{P}^1(\mathbb{Q}) \end{align*} be the extended upper half-plane. Let \begin{align*} \pi:\mathfrak{H}^* \to X_0(N) \end{align*} be the quotient map induced by the action of $\Gamma_0(N)$. The Farey tessellation decomposes $\mathfrak{H}$ into ideal hyperbolic triangles whose vertices lie in $\mathbb{P}^1(\mathbb{Q})$. Its oriented $1$-simplices are the oriented geodesic segments from $r$ to $s$, where $r,s\in \mathbb{P}^1(\mathbb{Q})$ are adjacent vertices in the tessellation. After applying $\pi$, each such oriented edge defines a relative singular $1$-chain in the pair \begin{align*} (X_0(N),\operatorname{Cusps}_0(N)). \end{align*} Its relative homology class is the modular symbol $\{r,s\}$. Because the Farey tessellation is $\Gamma_0(N)$-invariant, its quotient gives a relative cell decomposition of the pair $(X_0(N),\operatorname{Cusps}_0(N))$. Therefore the relative cellular chain group in degree $1$ is generated by the images of oriented geodesic edges between rational cusps. Passing from relative chains to relative homology preserves generation, so \begin{align*} H_1(X_0(N),\operatorname{Cusps}_0(N);\mathbb{Z}) \end{align*} is generated by the classes $\{r,s\}$ with $r,s\in \mathbb{P}^1(\mathbb{Q})$. [/step] [step:Parametrize all oriented Farey edges by cosets of $SL_2(\mathbb{Z})$] Let $e$ denote the oriented Farey edge from $0$ to $\infty$. The group $SL_2(\mathbb{Z})$ acts transitively on oriented Farey edges. Explicitly, if the oriented edge has initial vertex $s=b/d$ and terminal vertex $r=a/c$, then adjacency means \begin{align*} ad-bc=\pm 1. \end{align*} After replacing one column by its negative if necessary, choose \begin{align*} g=\begin{pmatrix}a&b\\ c&d\end{pmatrix}\in SL_2(\mathbb{Z}). \end{align*} Then $g(\infty)=a/c=r$ and $g(0)=b/d=s$, so the oriented edge from $s$ to $r$ is exactly the $g$-translate of $e$. Define \begin{align*} \Phi:\mathbb{Z}[\Gamma_0(N)\backslash SL_2(\mathbb{Z})] &\to H_1(X_0(N),\operatorname{Cusps}_0(N);\mathbb{Z})\\ [\Gamma_0(N)g] &\mapsto \{g(0),g(\infty)\}. \end{align*} This is well-defined because if $\gamma\in \Gamma_0(N)$, then the paths from $g(0)$ to $g(\infty)$ and from $\gamma g(0)$ to $\gamma g(\infty)$ have the same image in the quotient $X_0(N)$. By the generation proved in the previous step and the transitivity of the $SL_2(\mathbb{Z})$ action on Farey edges, $\Phi$ is surjective. [/step] [step:Derive the edge-reversal relation from the matrix $S$] The matrix \begin{align*} S=\begin{pmatrix}0&-1\\ 1&0\end{pmatrix} \end{align*} satisfies $S(0)=\infty$ and $S(\infty)=0$. Hence for every $g\in SL_2(\mathbb{Z})$, \begin{align*} gS(0)=g(\infty), \qquad gS(\infty)=g(0). \end{align*} Therefore \begin{align*} \Phi([\Gamma_0(N)gS]) = \{g(\infty),g(0)\}. \end{align*} Reversing the orientation of a relative $1$-chain multiplies its homology class by $-1$, so \begin{align*} \{g(\infty),g(0)\}=-\{g(0),g(\infty)\}. \end{align*} Thus \begin{align*} \Phi([\Gamma_0(N)g]+[\Gamma_0(N)gS])=0. \end{align*} This proves that every relation $[x]+[xS]$ lies in $\ker \Phi$. [/step] [step:Derive the triangle relation from the standard ideal triangle] The matrix \begin{align*} U=\begin{pmatrix}0&-1\\ 1&-1\end{pmatrix} \end{align*} acts on $\mathbb{P}^1(\mathbb{Q})$ by fractional linear transformations and satisfies \begin{align*} U(\infty)=0, \qquad U(0)=1, \qquad U^2(\infty)=1, \qquad U^2(0)=\infty. \end{align*} Thus the three Manin symbols associated to $g$, $gU$, and $gU^2$ are the oriented edges of the ideal triangle with vertices \begin{align*} g(0),\qquad g(\infty),\qquad g(1). \end{align*} More precisely, \begin{align*} \Phi([\Gamma_0(N)g]) &= \{g(0),g(\infty)\},\\ \Phi([\Gamma_0(N)gU]) &= \{g(1),g(0)\},\\ \Phi([\Gamma_0(N)gU^2]) &= \{g(\infty),g(1)\}. \end{align*} The boundary of the oriented ideal triangle with vertices $g(0)$, $g(\infty)$, and $g(1)$ is null in relative homology because it is the boundary of a relative $2$-chain. Therefore \begin{align*} \{g(0),g(\infty)\} + \{g(\infty),g(1)\} + \{g(1),g(0)\} =0. \end{align*} Reordering the three summands gives \begin{align*} \Phi([\Gamma_0(N)g]+[\Gamma_0(N)gU]+[\Gamma_0(N)gU^2])=0. \end{align*} Hence every relation $[x]+[xU]+[xU^2]$ lies in $\ker \Phi$. [/step] [step:Identify the kernel using the relative cellular boundary map] Let $C_1$ denote the free abelian group on oriented Farey edges modulo the orientation relation $[\bar e]+[e]=0$, where $\bar e$ is the same geometric edge with the opposite orientation. Let $C_2$ denote the free abelian group on oriented Farey triangles. The Farey tessellation of $\mathfrak{H}$ is an $SL_2(\mathbb{Z})$-equivariant contractible two-dimensional cell decomposition, and after quotienting by $\Gamma_0(N)$ it gives a relative cellular chain complex for the pair $(X_0(N),\operatorname{Cusps}_0(N))$: \begin{align*} C_2(\Gamma_0(N)\backslash \mathcal{T}) \xrightarrow{\partial_2} C_1(\Gamma_0(N)\backslash \mathcal{E}) \xrightarrow{\partial_1} C_0(\Gamma_0(N)\backslash \mathbb{P}^1(\mathbb{Q})) \to 0, \end{align*} where $\mathcal{T}$ is the set of oriented Farey triangles and $\mathcal{E}$ is the set of oriented Farey edges. Passing to relative chains with respect to the cusps kills $C_0(\Gamma_0(N)\backslash \mathbb{P}^1(\mathbb{Q}))$, so \begin{align*} H_1(X_0(N),\operatorname{Cusps}_0(N);\mathbb{Z}) \cong C_1(\Gamma_0(N)\backslash \mathcal{E}) / \operatorname{im}\partial_2. \end{align*} Thus all relations among the relative homology classes of oriented edges are generated by two sources: reversing the orientation of an edge, and taking the cellular boundary of an oriented Farey triangle. Under the parametrization of oriented edges by $\Gamma_0(N)\backslash SL_2(\mathbb{Z})$, the orientation reversal of the edge represented by $x$ is represented by $xS$. Hence the orientation relations are exactly \begin{align*} [x]+[xS], \qquad x\in \Gamma_0(N)\backslash SL_2(\mathbb{Z}). \end{align*} The standard oriented triangle has boundary represented by the three oriented edges attached to $1$, $U$, and $U^2$; translating by $g\in SL_2(\mathbb{Z})$ gives the boundary relation \begin{align*} [\Gamma_0(N)g]+[\Gamma_0(N)gU]+[\Gamma_0(N)gU^2]. \end{align*} Since every Farey triangle is an $SL_2(\mathbb{Z})$-translate of the standard one, these translated triangle relations generate all of $\operatorname{im}\partial_2$. Therefore \begin{align*} \ker \Phi = \left\langle [x]+[xS],\ [x]+[xU]+[xU^2] :\ x\in \Gamma_0(N)\backslash SL_2(\mathbb{Z}) \right\rangle. \end{align*} [/step] [step:Conclude the Manin-symbol presentation] Since $\Gamma_0(N)$ has finite index in $SL_2(\mathbb{Z})$, the set \begin{align*} \Gamma_0(N)\backslash SL_2(\mathbb{Z}) \end{align*} is finite. Therefore $\mathbb{Z}[\Gamma_0(N)\backslash SL_2(\mathbb{Z})]$ is a finitely generated free abelian group. The preceding step identifies the kernel of $\Phi$ exactly, not only a subgroup of the kernel. Hence $\Phi$ induces an isomorphism \begin{align*} \mathbb{Z}[\Gamma_0(N)\backslash SL_2(\mathbb{Z})]\Big/ \left\langle [x]+[xS],\ [x]+[xU]+[xU^2] :\ x\in \Gamma_0(N)\backslash SL_2(\mathbb{Z}) \right\rangle &\xrightarrow{\ \cong\ } H_1(X_0(N),\operatorname{Cusps}_0(N);\mathbb{Z}),\\ [x] &\mapsto \{g(0),g(\infty)\}, \end{align*} where $x=\Gamma_0(N)g$. This is a finite presentation because the coset set is finite and the displayed $S$ and $U$ relations are finite in number over that coset set. This proves both the generation by modular symbols and the finite Manin-symbol presentation. [/step]