Topological Collapse Separability Theorem

• Topological Collapse Separability Theorem is a unifying framework that defines equivalences between discrete simplicial collapsibility, deformation retractions in PL topology, and separability transitions in quantum many-body systems.
• The theorem rigorously demonstrates that a PL polyhedron collapses to a subpolyhedron if and only if there exists a PL free deformation retraction, linking combinatorial operations with homotopical properties.
• In quantum settings, the theorem identifies the separability threshold as the critical point where local decoherence transforms long-range entangled states into short-range entangled mixtures, aligning with classical phase transitions.

The Topological Collapse Separability Theorem unifies several distinct frameworks—piecewise-linear (PL) topology, real singularity theory, and quantum information—through the concept that critical structural or entanglement transitions are captured by equivalences between topological operations (collapses or separating sets), deformation retractions, and separability thresholds. In PL topology, the theorem rigorously characterizes when a polyhedron simplicially collapses onto a subpolyhedron via the existence of a specific class of deformation retractions. In the topology of singularities, it establishes an equivalence between geometric conditions (separating sets), tangent link behavior, and nontrivial fast-contracting homology in the thin zone. In quantum many-body systems, the theorem equates the transition from topological to short-range entangled mixed states under local decoherence with a phase transition in a classical statistical model, locating the separability threshold at the point of error-correction breakdown.

1. Definitions and Core Statements

In the PL category, the Topological Collapse Separability Theorem states that for a compact PL-polyhedron $P$ and subpolyhedron $Q \subset P$, the following are equivalent:

1. There exists a piecewise-linear free deformation retraction $f: P \times I \to P$ of $P$ onto $Q$, with
   • $f(x, 0) = x$ for all $x \in P$
   • $f(P, 1) = Q$
   • $f(q, t) = q$ for all $q \in Q$, $t \in I$,
   • and the freeness property: $f(f(x, s), t) = f(x, \max\{s, t\})$.
2. $P$ simplicially collapses to $Q$—i.e., admits a triangulation such that $T$ (the triangulation of $P$) can be reduced to $T_Q$ (the subcomplex triangulating $Q$) by a finite sequence of elementary simplicial collapses (Gorelov, 2021).

In singularity theory, the theorem applies to closed definable thick isolated singularity germs $(X, 0) \subset (\mathbb{R}^n, 0)$ with connected link, using the decomposition into thick and thin zones. There, the following are equivalent:

• The existence of a separating set in $X$ (a codimension-1, thin-at-0, closed definable subset such that $(X \setminus S, 0)$ has at least two thick connected components).
• Non-simple tangent directions in the tangent link $S_{\textrm{ns}} X \subset S^{d-1}$ satisfy a certain separation condition (SC).
• The inclusion

$$\iota_*: H_{d-2}(\mathrm{Thin}(X), 0) \to H_{d-2}(X, 0)$$

on fast-contracting homology is surjective but not injective; specifically, $\ker \iota_* \neq 0$ (Birbrair et al., 2012).

In topological phases of quantum matter, the theorem manifests as a separability transition: for a CSS-type topologically ordered state $\rho_0$ subjected to local (bit- or phase-flip) errors with probability $p$, the decohered mixed state $\rho(p)$ can be expressed as a convex sum of short-range entangled (SRE) pure states (i.e., is SRE-separable) if and only if $p > p_c$, where $p_c$ coincides with the corresponding classical transition point (e.g., on the Nishimori line). For $p < p_c$, $\rho(p)$ retains long-range entanglement and cannot be so decomposed (Chen et al., 2023).

2. Piecewise-Linear Collapsibility and Free Deformation Retraction

An elementary simplicial collapse removes a pair $(A, B)$, where $A$ is a simplex of a finite simplicial complex $K$ and $B \subset A$ is a codimension-1 face not shared with any other simplex. $K$ simplicially collapses to a subcomplex $K_Q$ if there is a finite sequence of such collapses taking $K$ to $K_Q$.

A piecewise-linear free deformation retraction (PL-FDR) $f: P \times I \to P$ is a PL map satisfying the strong deformation retraction and the additional "freeness" property: composing the retraction at different times is governed by the maximum—$f_t \circ f_s = f_{\max(t, s)}$.

The main theorem (Gorelov, 2021) provides a combinatorial-homotopical correspondence:

Polyhedral Operation	Homotopical Analog
Simplicial collapse from $P$ to $Q$	Existence of PL-FDR $P \to Q$

Proof proceeds in both directions: Simplicial collapse sequences can be realized stepwise as explicit PL-FDRs by induction and gluing (using a lemma enabling concatenation of free retractions); conversely, a PL-FDR yields an upward-closed subcomplex whose cylindrical triangulation supports a collapse sequence projecting down to $P \to Q$. This result links discrete (simplicial) combinatorics to the PL-homotopy category, giving a homotopy-invariant characterization of the combinatorial notion of collapsibility.

Corollaries include equivalence for 2-polyhedra (Isbell) and 3-manifolds with boundary (Piergallini), and the demonstration that certain high-dimensional contractible non-collapsible spaces lack PL-FDRs, positioning the theorem as a sharp diagnostic for PL collapsibility.

3. Separating Sets and Fast-Contracting Homology in Singularities

In real definable singularities, the thick/thin decomposition partitions a germ into regions with simple versus non-simple tangent links. The topological thin zone $\operatorname{Thin}(X)$ is defined by neighborhoods of the non-simple locus $S_{\rm ns} X$ in the strict transform after spherical blow-up.

A separating set is a codimension-1 thin-at-0 subset whose tangent cone separates the ambient tangent cone at the origin, resulting in multiple thick components.

Fast contracting homology captures cycles in the links of $X$ that bound chains supported in the thin zone and contract "faster than linearly." The map $\iota_*$ encodes whether such cycles survive when "collapsed" to the thick/thin decomposition.

The equivalence (Birbrair et al., 2012):

Geometric/Topological	Tangent Link Condition	Homological Obstruction
Existence of separating set	$S_{\rm ns} X$ satisfies (SC)	Ker$(\iota_*) \neq 0$ on $H_{d-2}$

All known obstructions to metric conicalness (fast loops, choking horns, separating sets) are realized by nontrivial fast-contracting classes in the thin zone. Morphisms (so-called rigid homeomorphisms) preserve this decomposition and the associated homology, showing the stability of this obstruction under strong topological equivalence.

4. Mixed-State Separability Transitions in Topological Quantum Phases

For CSS-type topologically ordered codes, subjecting the system to independent local decoherence translates the problem of separability (convex decomposition into SRE pure states) into a statistical transition governed by a classical model (typically, a random-bond or random-plaquette Ising model on the Nishimori line).

Given the decohered state

$$\rho(p) = \mathbb{E}_p[\rho_0] = \bigotimes_e \big[ p E_e \rho_0 E_e + (1 - p) \rho_0 \big],$$

the state $\rho(p)$ is SRE if and only if $p > p_c$, with $p_c$ determined by the classical phase transition. The spectral-square-root construction produces a decomposition

$$\rho(p) = \sum_g w_g |\psi_g\rangle\langle \psi_g|,$$

where each $|\psi_g\rangle$ is SRE for $p > p_c$. The threshold $p_c$ aligns precisely with the optimal error-correction threshold for the corresponding code (e.g., 2D toric code: $p_c \approx 0.109$; 3D phase-flip: $p_c \approx 0.029$; X-cube fracton: $p_c \approx 0.152$) (Chen et al., 2023).

Table of models and separability thresholds:

Model	Error Map & Classical Model	$p_c$
2D toric code, bit-flip	2D random-bond Ising, Nishimori line	$\approx$ 0.109
3D toric code, phase-flip	3D random-plaquette gauge model	$\approx$ 0.029
3D toric code, bit-flip	3D random-bond Ising	$\approx$ 0.233
X-cube fracton, phase-flip	3D plaquette-Ising (Savvidy–Wegner)	$\approx$ 0.152

This construction identifies the many-body separability transition as a "collapse" of topological order, precisely coinciding with failure of active error correction.

5. Illustrative Examples and Physical Interpretation

PL Topology: In the case $Q$ is a point and $P$ is a compact 2-polyhedron, the theorem specializes to the criterion for collapsibility to a point. High-dimensional analogues illustrate the necessity of PL-FDR for combinatorial collapsibility, with the Zeeman dunce-hat as a key open problem.

Singularity Theory: For the Pham–Brieskorn surface $\{x^a + y^b + z^c = 0\}$ in $\mathbb{C}^3$ (with suitable $a, b, c$), the thin zone is the cone over a real codimension-1 torus of non-simple tangents; separating sets correspond to nontrivial fast-contracting cycles. Visualization in $\mathbb{R}^3$ with thin zone as the cone over an "equatorial belt" clarifies the geometric mechanism of separation.

Quantum Phases: The LRE-to-SRE transition in noisy toric codes is witnessed by the vanishing of topological order parameters (Wilson loops, 't Hooft loops) and the explicit SRE decomposition of the mixed state above threshold, unifying quantum error correction with many-body entanglement transitions.

6. Generalizations, Applications, and Limitations

The theorem's structure is robust under several generalizations:

• In singularity theory, the characterization extends to any closed isolated singularity definable in a polynomially-bounded o-minimal expansion of $\mathbb{R}$, potentially to subanalytic/log-analytic categories given sufficient control on blow-ups (Birbrair et al., 2012).
• Higher-codimension separating loci yield analogous statements for higher-degree homology.
• In quantum settings, applicability is direct for all CSS stabilizer codes with cluster-state parents; non-CSS or non-Abelian codes require extensions of the formalism (Chen et al., 2023).
• The bi-Lipschitz invariance of the fast-contracting homology of the thin zone yields stable topological obstructions across a wide class of homeomorphisms.

The methods deployed—cylindrical triangulations, shadow projections, upward and downward closures—serve as templates for translating between continuous homotopical data and combinatorial operations, potentially informing the study of wild PL topologies, obstruction theory, and complex group actions on high-dimensional CW complexes.

7. Scope, Significance, and Open Problems

The Topological Collapse Separability Theorem constitutes a conceptual nexus across PL topology, singularity theory, and quantum information. It delivers characterization theorems at the intersection of combinatorics, topology, and statistical mechanics, revealing that discrete collapse processes, contractible spectra, and geometric singularities are unified under questions of collapse, separation, and homological triviality in thin zones. This synthesis facilitates quantitative and structural diagnostics for topological complexity, metric conicalness obstructions, and the stability of quantum topological order under local decoherence.

Open questions include the full characterization of collapse in complex singularities across broader o-minimal settings and the extension of separability transitions to non-CSS or non-Abelian quantum codes. The Zeeman dunce-hat conjecture remains unresolved as a touchstone for the relation between contractibility and PL-collapsibility (Gorelov, 2021). The universality of these principles for mixed-state topology in quantum systems hints at further deep links between topological invariants and operationally meaningful criteria in quantum error correction and phase classification.