= Solution
A useful closed-cover version of the <Seifert-van Kampen theorem> is the following. Let $X=A\cup B$ with $A,B,A\cap B$ path-connected closed subspaces and basepoint in their intersection. Assume there are open neighbourhoods $U,V$ covering $X$ that deformation retract onto $A,B$, while $U\cap V$ deformation retracts onto $A\cap B$, with the induced inclusions compatible up to based homotopy. Then
$$
\pi_1(X)=\pi_1(A)*\pi_1(B)/\langle\!\langle i_A(\gamma)i_B(\gamma)^{-1}:\gamma\in\pi_1(A\cap B)\rangle\!\rangle.
$$
The neighbourhood hypothesis, available for the present finite subcomplex gluing, matters: arbitrary closed covers need not satisfy this theorem.
A <Möbius band> retracts onto its core circle, with generator $m$; its boundary represents $m^2$. The <torus> has generators $u,v$ and relation $[u,v]=1$. The attaching circle is the $u$ circle. Reversing the attaching homeomorphism merely replaces $u$ by its inverse and changes no isomorphism type. The closed-cover theorem gives
$$
\boxed{\pi_1(X)=\langle m,u,v\mid [u,v]=1,\ u=m^2\rangle
=\langle m,v\mid[m^2,v]=1\rangle}.
$$
Mapping $m\mapsto0$, $v\mapsto1$ surjects onto $\mathbb Z$, so the group is infinite. Imposing the extra relation $m^2=1$ gives the quotient $C_2*\mathbb Z$, a nonabelian <free product>, since alternating reduced words $mv$ and $vm$ differ. A quotient of an abelian group would be abelian, so \b[the <fundamental group> is nonabelian].
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