Boundary restriction and the surgery detector

Inclusion of the boundary torus into the exterior gives a homomorphism of fundamental groups. Precomposing a representation with this homomorphism is polynomial in matrix entries, so it is a regular map of representation varieties and is equivariant for simultaneous conjugation.

Write the attached solid torus as $V=D^2\times S^1$, let $A=\{0\}\times S^1$, and base at $(0,1)$. The retraction $r(z,w)=(0,w)$ and inclusion $i:A\hookrightarrow V$ satisfy $r\circ i={\mathrm{id}}_A$, and the homotopy $((z,w),t)\mapsto ((1-t)z,w)$ from ${\mathrm{id}}_V$ to $i\circ r$ fixes the basepoint. By [F5] and [F6], ${\pi }_1(V,(0,1))\cong {\pi }_1(S^1,1)\cong \mathbb{Z}$.

By [A1] and [F7], the exterior boundary has a collar; $V$ has its product collar. Thickening the two pieces across opposite halves of the gluing collar gives an open cover of the filled manifold whose sets retract to $E_K$ and $V$ and whose intersection retracts to the torus. These spaces are path connected, so [F4] identifies the filled fundamental group as their pushout.

Pullback along this equivariant map sends conjugation-invariant regular functions to conjugation-invariant regular functions, hence induces the algebraic restriction morphism on affine categorical quotients.

Since $\mu ,\lambda$ commute, the primitive class $p[\mu ]+q[\lambda ]$ is represented by ${\mu }^p{\lambda }^q$. It maps to the identity in ${\pi }_1(V)$ because it is the meridian. Complete $(p,q)$ to a unimodular basis of ${\mathbb{Z}}^2$ by Bézout; the complementary class has longitude coefficient $\pm 1$ on $\mathrm{\partial }V$, hence maps to a generator of ${\pi }_1(V)$ by step 1.2. The pushout from step 1.3 identifies this generator with the image of the complementary class in ${\pi }_1(E_K)$ and imposes only the relation ${\mu }^p{\lambda }^q=1$. Eliminating the generator gives ${\pi }_1(E_K(p\mathrm{/}q))\cong {\pi }_1(E_K)\mathrm{/}⟨\text{\hspace{0.17em}⁣}⟨{\mu }^p{\lambda }^q⟩\text{\hspace{0.17em}⁣}⟩$.

By [F2], the defining equation of the reduced curve union vanishes on each of its one-dimensional image components. The definition takes only those components, so an isolated image point outside their union is not added as a curve factor.

A homomorphism from this quotient is exactly a homomorphism from ${\pi }_1(E_K)$ that kills ${\mu }^p{\lambda }^q$. Since $\rho (\mu )$ and $\rho (\lambda )$ commute, this condition is the matrix equation $\rho (\mu {)}^p\rho (\lambda {)}^q=I_2$.

If the two peripheral matrices are simultaneously diagonalized, their product of powers is $\mathrm{diag}(M^p{L}^q,(M^p{L}^q{)}^{-1})$, which is the identity exactly when $M^p{L}^q=1$. For a merely triangular pair, the diagonal entries still give this necessary equation, but they do not determine the off-diagonal entries of the product; the matrix relation remains the exact test.