%
\label{thm:fundamental-group-degeneration}
The pinch map induces a surjection on fundamental groups:
\[
    \boxed{%
    \pi_1(T^2)
    \;=\;
    \mathbb{Z}^2
    \;\xrightarrow{\quad p_* \quad}\;
    \pi_1(\mathbb{L})
    \;=\;
    F_2,}
\]
where $F_2$ is the free group
on generators $a$ ($+$-lobe loop)
and $b$ ($-$-lobe loop).

$\pi_1(T^2) = \mathbb{Z} \times \mathbb{Z}$,
generated by coordinate loops
$\gamma_0 : t \mapsto (t, 0)$
and $\eta_0 : t \mapsto (0, t)$,
which commute:
$[\gamma_0] \cdot [\eta_0]
= [\eta_0] \cdot [\gamma_0]$.
By van~Kampen,
$\pi_1(S^1 \vee S^1) = F_2$
with generators $a = [p_*(\gamma_0)]$
and $b = [p_*(\eta_0)]$
that do \emph{not} commute: $ab \neq ba$.
The pinch map sends the abelian generators
to free generators;
the commutativity relation ceases to hold.
Surjectivity follows because $\gamma_0, \eta_0$
generate $\pi_1(T^2)$.
The relationship is:
$F_2^{\mathrm{ab}} = \mathbb{Z}^2$---the
abelianization map run in reverse.