Why Topology? Necessity from “Contractible” to “Punctured”

In previous section, we saw topological constraint $[K]$ plays key role in unified theory. But a natural question is: Why do we need relative topology? Why can’t we directly use absolute topological invariants?

Answer lies hidden in a mathematical fact: Full-rank density matrix manifold is open convex contractible.

“Topological Triviality” of Contractible Spaces

What Is Contractible Space?

A topological space $X$ is called contractible if it can continuously “shrink” to a point, mathematically meaning identity map ${\mathrm{id}}_X$ is homotopic to constant map.

Key Fact: Contractible space is homotopy equivalent to a point, therefore: $$H^n(X)=\{\begin{array}{ll}\mathbb{Z}& n=0\\ 0& n\ge 1\end{array}$$

In other words, contractible spaces have no non-trivial absolute cohomology classes!

Contractibility of Density Matrix Manifold

Consider $N$-dimensional full-rank density matrix manifold: $${\mathcal{D}}_N^{\mathrm{full}}=\{\rho \in {\mathrm{Herm}}_N^{+}:\rho >0,\mathrm{tr}\rho =1\}$$

Theorem: ${\mathcal{D}}_N^{\mathrm{full}}$ is contractible.

Proof Idea: Construct explicit contraction map $$H_t(\rho )=(1-t)\rho +t\cdot \frac{I}{N},t\in [0,1]$$

• When $t=0$: $H_0(\rho )=\rho$ (identity map)
• When $t=1$: $H_1(\rho )=I/N$ (maximally mixed state, fixed point)
• For all $t\in [0,1]$: $H_t(\rho )\in {\mathcal{D}}_N^{\mathrm{full}}$ (convex combination preserves positivity)

graph LR
    A["Arbitrary State ρ"] -->|t=0| A
    A -->|"t∈(0,1)"| B["Mixed State Path<br/>(1-t)ρ+tI/N"]
    B -->|t=1| C["Maximally Mixed State<br/>I/N"]

    style A fill:#d4a5a5
    style B fill:#ffd93d
    style C fill:#6bcf7f

Physical Meaning: Any quantum state can continuously “decohere” to completely mixed state by mixing with maximally mixed state.

Catastrophic Consequences of Contractibility

Because ${\mathcal{D}}_N^{\mathrm{full}}$ is contractible, we have:

$$H^2({\mathcal{D}}_N^{\mathrm{full}};{\mathbb{Z}}_2)=0$$

This means:

• No non-trivial absolute cohomology classes
• No integer-valued topological invariants
• Cannot distinguish different physical phases using absolute topology

This is a topological vacuum—all states are topologically indistinguishable!

graph TD
    A["Full-Rank Density Matrix Manifold<br/>D_N^full"] -->|Contractibility| B["H²(D_N^full)=0"]

    B --> C["Absolute Topological Vacuum"]
    C --> D["No Integer Invariants"]
    C --> E["No Phase Distinction"]
    C --> F["No Topological Constraints"]

    style B fill:#ff6b6b
    style C fill:#d4a5a5

Global Section of Uhlmann Principal Bundle

Another manifestation of contractibility: Uhlmann principal bundle on full domain admits global square root section.

Uhlmann principal bundle is defined as: $$P=\{w=\sqrt{\rho }U:\rho \in {\mathcal{D}}_N^{\mathrm{full}},U\in U(N)\}$$ Projection map: $$\pi (w)=w{w}^{†}=\rho$$

On full domain, we can always choose a globally continuous square root section: $$\sigma (\rho )=\sqrt{\rho }$$

where $\sqrt{\rho }$ is positive definite Hermitian square root (unique and smooth).

Topological Meaning: Principal bundle is trivial, has no non-trivial characteristic classes!

Puncturing: Necessary Surgery to Break Contractibility

Physical Motivation: Inevitability of Degeneracy

In physical processes, certain special configurations are inevitable:

1. Level Degeneracy: Two or more eigenvalues equal
2. Phase Transition Points: Boundaries between physical phases
3. Singularities: Certain physical quantities diverge

These special points form discriminant locus, places that are physically important but topologically “pathological”.

Removing Discriminant Set: Puncturing Operation

For $N=5$ case, consider three-two level degeneracy set: $${\Sigma }_{3|2}=\{\rho \in {\mathcal{D}}_5^{\mathrm{full}}:{\lambda }_3={\lambda }_4\}$$

Here ${\lambda }_1\ge {\lambda }_2\ge {\lambda }_3\ge {\lambda }_4\ge {\lambda }_5>0$ are eigenvalues.

Puncturing Operation: Remove tubular neighborhood of ${\Sigma }_{3|2}$ $${\mathcal{D}}^{\mathrm{exc}}=\{\rho \in {\mathcal{D}}_5^{\mathrm{full}}:g(\rho )\ge 2\delta \}$$ where spectral gap function $g(\rho )={\lambda }_3-{\lambda }_4$.

graph TD
    A["Full Domain D⁵_full<br/>Contractible Space"] --> B["Degeneracy Set Σ₃|₂<br/>λ₃=λ₄"]

    A -.->|Remove| C["Punctured Domain D^exc<br/>λ₃-λ₄≥2δ"]

    C --> D["Non-Contractible"]
    D --> E["Non-Trivial Topology"]
    E --> F["Relative Cohomology Class [K]"]

    style B fill:#ff6b6b
    style C fill:#ffd93d
    style F fill:#6bcf7f

Geometry of Degeneracy Set: Codimension 3 and S² Link

Proposition: In three-dimensional transverse slice maintaining open gap $({\lambda }_2,{\lambda }_5)$ and no additional symmetry, ${\Sigma }_{3|2}$ is codimension 3 normal subset, whose small ball boundary link is homotopic to $S^2$.

Proof Idea:

1. Restrict Hamiltonian to near-degenerate 2D eigenspace
2. Get $h=x{\sigma }_x+y{\sigma }_y+z{\sigma }_z$
3. Degeneracy condition: $(x,y,z)=(0,0,0)$ (three independent real constraints)
4. Take normal small ball $B^3$, its boundary $S^2$ is link

Physical Meaning:

• Degeneracy points are isolated cone points in parameter space
• Small loops around degeneracy points are topologically non-trivial
• $S^2$ link encodes topological charge of degeneracy

graph LR
    A["3D Parameter Space<br/>(x,y,z)"] --> B["Degeneracy Point<br/>(0,0,0)"]
    B --> C["Codimension 3 Cone Point"]

    C --> D["Small Ball B³"]
    D --> E["Boundary Link<br/>S²"]

    E --> F["Topological Charge<br/>Non-Trivial π₂"]

    style B fill:#ff6b6b
    style E fill:#6bcf7f

Birth of Relative Topology

Pair Space $(Y,\partial Y)$

After puncturing, we get a pair space: $$Y={\mathcal{D}}^{\mathrm{exc}},\partial Y=\partial {\mathrm{Tub}}_{\epsilon }({\Sigma }_{3|2})$$

• $Y$: Punctured domain (interior)
• $\partial Y$: Tubular neighborhood boundary

Long Exact Sequence of Relative Cohomology

Pair space $(Y,\partial Y)$ induces long exact sequence of relative cohomology: $$\cdots \to H^1(Y)\to H^1(\partial Y)\stackrel{\partial }{\to }{H}^2(Y,\partial Y)\to H^2(Y)\to H^2(\partial Y)\to \cdots$$

Key Observations:

1. $H^2(Y)=0$ ($Y$ still “almost contractible”)
2. $H^2(\partial Y)\ne 0$ (boundary has non-trivial topology)
3. $H^2(Y,\partial Y)\ne 0$ (relative classes exist!)

Boundary map $\partial :H^1(\partial Y)\to H^2(Y,\partial Y)$ is source of relative classes.

Physical Meaning of Relative Classes

Relative cohomology class $[K]\in H^2(Y,\partial Y;{\mathbb{Z}}_2)$ encodes:

1. Topological Memory of Boundary Conditions

   • On boundary $\partial Y$, scattering matrix may have phase jumps
   • These jumps “integrate” along interior paths into topological charges

2. Topological Shadow of Discriminant Set

   • Although ${\Sigma }_{3|2}$ is removed, its “trace” remains in relative classes
   • Loops around discriminant set leave ℤ₂ imprint on $\partial Y$

3. Realizability Constraints of Physical Processes

   • If $[K]\ne 0$, certain quantum processes accumulate non-eliminable phases on loops
   • This phase causes topological destruction of quantum interference

Why Must We Puncture? Four Levels of Reasons

Reason 1: Mathematical Necessity

Contractibility Theorem: Full domain ${\mathcal{D}}_N^{\mathrm{full}}$ has no non-trivial absolute topological classes.

To obtain topological constraints, must break contractibility, only method is to remove certain subsets.

Reason 2: Physical Regularity

At degeneracy points, many physical quantities are non-regular:

• Riesz spectral projection discontinuous
• Berry phase may diverge
• Adiabatic approximation fails

Removing degeneracy set ensures physical processes are smooth and regular on ${\mathcal{D}}^{\mathrm{exc}}$.

Reason 3: Topological Quantization

Only on punctured domain can we define:

• Unified Contour Family: Family of closed curves maintaining finite distance from co-spectrum
• Riesz Projection: $P_3(\rho )=\frac{1}{2\pi i}{\oint }_{{\gamma }_3}(z-\rho {)}^{-1}dz$
• Principal Bundle Reduction: $U(5)\to U(3)\times U(2)$

These constructions become singular at degeneracy points.

Reason 4: Categorical Terminal Object Property

In terminal object construction of unified theory, $[K]=0$ is consistency axiom. If $[K]\ne 0$, it causes:

• Breaking of scattering-modular flow alignment
• Incompatibility of boundary time and geometric time
• Sign flip of generalized entropy variation

Puncturing and requiring $[K]=0$ is forced by self-consistency.

graph TD
    A["Contractible Full Domain<br/>No Absolute Topology"] --> B["Must Puncture"]

    C["Physical Regularity"] --> B
    D["Topological Quantization"] --> B
    E["Terminal Object Consistency"] --> B

    B --> F["Punctured Domain D^exc"]
    F --> G["Relative Cohomology [K]"]
    G --> H["Topological Constraint<br/>[K]=0"]

    style A fill:#d4a5a5
    style B fill:#ff6b6b
    style G fill:#ffd93d
    style H fill:#6bcf7f

Analogy: Singularity Removal on Smooth Manifolds

Consider a physical analogy: Point charge in electromagnetism.

Absolute Case: Including Singularity

If we consider point charge on full space ${\mathbb{R}}^3$ (including origin):

• Potential $\varphi =1/r$ is singular at origin
• Electric field $\mathbf{E}=-\nabla \varphi$ diverges at origin
• Gauss theorem on any closed surface containing origin gives non-zero flux

But if we ask: “What are topological properties of ${\mathbb{R}}^3$?”

• Answer: ${\mathbb{R}}^3$ is contractible, $H^1({\mathbb{R}}^3)=0$
• Vector potential $\mathbf{A}$ can be globally defined (but singular at origin)

Relative Case: Punctured Space

If we remove origin, get ${\mathbb{R}}^3\setminus \{0\}$:

• Topologically non-trivial: $H^1({\mathbb{R}}^3\setminus \{0\})=0$ but $H^2({\mathbb{R}}^3\setminus \{0\})\ne 0$ (through Poincaré duality)
• Boundary $\partial (\mathrm{Tub}(0))=S^2$ (small sphere)
• Relative class encodes magnetic flux through $S^2$

This is exactly Dirac string construction of magnetic monopole! After removing singularity line, topological charge of magnetic monopole appears as relative class in $H^2$.

graph LR
    A["R³<br/>Contractible"] -.->|Remove Origin| B["R³\{0}<br/>Punctured"]

    A --> C["No Topological Charge"]
    B --> D["Boundary S²"]
    D --> E["Relative Class<br/>Magnetic Flux"]

    style A fill:#d4a5a5
    style B fill:#ffd93d
    style E fill:#6bcf7f

Summary: Paradigm Shift from Absolute to Relative

Core Insight:

Puncturing is not “technical patch”, but inevitable requirement of topological quantization.

Just as magnetic monopole needs to remove Dirac string to define magnetic charge, density matrix manifold needs to remove degeneracy set to define topological constraints.

Next Step: Precise Definition of Relative Cohomology Class [K]

Now we understand why relative topology is needed. Next section will give precise mathematical definition of $[K]$ and analyze physical meaning of its three-term decomposition:

$$[K]={\pi }_M^{\ast }{w}_2(TM)+\sum _j{\pi }_M^{\ast }{\mu }_j⌣{\pi }_X^{\ast }{\mathfrak{w}}_j+{\pi }_X^{\ast }\rho (c_1({\mathcal{L}}_S))$$

Each term has profound physical origin!