Pinch Map

The surjection p: T^2 -> L collapsing the diagonal circle to the wedge point. The canonical degeneration of torus to lemniscate.

Pinch Map

Summary

The surjection p: T^2 -> L collapsing the diagonal circle to the wedge point. The canonical degeneration of torus to lemniscate.

Statement

%
\label{def:pinch-map}
The \textbf{pinch map} is the surjection
\[
    \boxed{p \;:\; T^2 = S^1_\gamma \times S^1_\eta
    \;\longrightarrow\;
    \mathbb{L} = S^1 \vee S^1}
\]
defined by collapsing the \textbf{diagonal circle}
$\Delta := \{(\theta, \theta) : \theta \in S^1\} \subset T^2$
to a single point~$\ast$, the \textbf{wedge point}:
\[
    p(\theta_\gamma, \theta_\eta)
    \;=\;
    \begin{cases}
        \ast
        & \text{if } \theta_\gamma = \theta_\eta, \\[3pt]
        [\theta_\gamma, \theta_\eta]
        & \text{if } \theta_\gamma \neq \theta_\eta,
    \end{cases}
\]
where $[\theta_\gamma, \theta_\eta]$
denotes the equivalence class in $T^2/\!\sim$.
The image $\mathbb{L} := T^2/\Delta$
is the \textbf{geometric lemniscate}.

Proof / Justification

This item is definitional. No manuscript proof is required.

Source Context

• Registry source: book-02.jsonl line 43
• Manuscript source: 2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part03/ch17-torus-degeneration.tex lines 128-154

Lean / Formalization Notes

• Formalization: formalized
• Module: TauLib.BookII.Topology.TorusDegeneration
• Name: pinch_fiber

Dependencies

• Canonical: II.D06, I.D18

Related Results

Generated by later projection phases.

Related Publications

Generated by later projection phases.

Revision Notes

• 2026-04-24: Initial pilot migration.