Independent meridian classes detected by individual winding maps

Parametrize each boundary loop as the singular one-simplex $c_k(t)={\gamma }_k(2\pi t)$ for $0\le t\le 1$. Its endpoints agree, so ${\mathrm{\partial }}_1{c}_k=c_k(1)-c_k(0)=0$. Thus $[{\gamma }_k]$ denotes precisely the class of $c_k$ in the cycle quotient of [F3].

Put $\eta (t)=e^{2\pi it}$. The circle is path-connected: writing a point as $e^{i\theta }$, the path $s\mapsto e^{is\theta }$ joins $1$ to it. By [F4], ${\pi }_1(S^1,1)\cong \mathbb{Z}$ with generator represented by $\eta$. This group is already abelian, so the degree-one Hurewicz isomorphism of [F4] sends that generator to the singular class $[\eta ]$. Consequently there is an isomorphism $\lambda :H_1(S^1;\mathbb{Z})\to \mathbb{Z}$ specified by $\lambda ([\eta ])=1$.

For each integer $\mathrm{\ell }$, restriction of $L_{\mathrm{\ell }}$ gives a continuous map $M\to {\mathbb{C}}^{\ast }$ by [F1]. Define the homomorphism $w_{\mathrm{\ell }}=\lambda \circ (r\circ L_{\mathrm{\ell }}{)}_{\ast }:H_1(M;\mathbb{Z})\to \mathbb{Z}$ using [F2]. Postcomposition on singular simplices means that it sends $[c_k]$ to $\lambda ([r\circ L_{\mathrm{\ell }}\circ c_k])$.

If $k=\mathrm{\ell }$, [F1] gives $L_{\mathrm{\ell }}(c_k(t))=\epsilon e^{2\pi it}$, so radial projection gives $\eta$ and $w_{\mathrm{\ell }}([{\gamma }_k])=1$. If $k\mathrm{\ne }\mathrm{\ell }$, $L_{\mathrm{\ell }}$ is nonzero on all of $D_k$. The extension $r\circ L_{\mathrm{\ell }}:D_k\to S^1$ contracts the boundary map $S^1\to S^1$ to a constant by contracting the disk linearly to its centre. By [F2] its induced image of $[\eta ]$ equals that of the constant map. A constant singular one-simplex is the boundary of the constant singular two-simplex, whose alternating faces have coefficients $1,-1,1$, so this image is zero by [F3]. Hence $w_{\mathrm{\ell }}([{\gamma }_k])$ is the Kronecker delta.

Apply $w_{\mathrm{\ell }}$ to any finite integral relation ${\sum }_k{n}_k[{\gamma }_k]=0$. Step 3.1 gives $n_{\mathrm{\ell }}=0$ for every $\mathrm{\ell }$, proving independence. The homomorphism from the finite-support direct sum ${⨁}_{k\in \mathbb{Z}}\mathbb{Z}$ that sends its $k$th basis element to $[{\gamma }_k]$ is therefore injective, giving the asserted subgroup.