Two Model Constructions for Stratified Vector Bundles on Singular Spaces

This post gives two concrete examples of stratified vector bundles — objects that are known to exist abstractly but rarely written down in explicit geometric form. Each example is smooth on its strata, continuous across singular boundaries, and fails to extend as an ordinary bundle on the full space. Together they offer the first geometric reference models for what nontrivial elements of stratified K-theory (Ks) actually look like.

Example 1 – Cone over a Circle (Monodromy Family)

Let X be the cone on a circle. The regular part looks like (0,∞) × S1, and the singular part is just the apex point.
For each integer m, define a stratified complex line bundle Em over X as follows:

• On the regular part, Em is a flat line bundle with transition map exp(i m θ) around the circle.
• On the apex, the fiber is set to 0 (the rank drops along the singular point).

Each Em is smooth on the regular stratum and continuous in the stratified sense. Two bundles Em and En are isomorphic only when m = n.

These bundles show that Ks of the cone on S1 detects both rank and monodromy. Holonomy around the apex produces distinct classes in Ks(X), so Ks(C(S1)) is expected to look like Z ⊕ Z — one copy for rank, one for degree.

Example 2 – Whitney Umbrella (Ruling Direction Bundle)

Consider the surface in R3 given by x² = y²z. Its singular set is the line where y = 0 and x = 0, while the rest of the surface is smooth.

On the smooth part, define a line subbundle L by taking at each point (u,v) the span of (v,1,0) — this is the ruling direction of one sheet of the umbrella.
As v approaches 0 (the singular line), this direction tends to (0,1,0).
Define the fiber over the singular line to be the span of (0,1,0).

This gives a rank-1 stratified line bundle that is smooth on both strata and continuous across them. It cannot be extended to an ordinary bundle globally because the umbrella self-intersects. The stratification makes the extension legal and well-defined.

Non-Explicit Families

1. Cone family: For any compact manifold Y, the cone C(Y) supports stratified bundles associated to unitary representations of the fundamental group of Y.
2. Whitney family: Any Whitney-stratified complex hypersurface with a one-dimensional singular locus has a limiting tangent-plane field that defines a natural stratified subbundle of the ambient tangent bundle.

These families generalize the two examples above and show that stratified bundles naturally arise from holonomy or from limiting tangent geometry.

How to Verify the Constructions

1. Check the stratification:
   • Cone: regular stratum (0,∞)×S1, singular apex point.
   • Umbrella: regular surface plus singular line.
     Both satisfy the frontier condition (closures of higher strata contain lower ones).
2. Smoothness on strata:
   • The map exp(i m θ) and the vector field (v,1,0) are smooth on their respective strata.
3. Continuity across strata:
   • For the cone, the classifying map into BU(1) extends continuously by sending the apex to the zero-rank point.
   • For the umbrella, (v,1,0) approaches (0,1,0) continuously as v→0.
4. Bundle compatibility:
   • Transition maps glue smoothly on each stratum and continuously across boundaries.
5. Classifying map picture:
   • Each stratified bundle corresponds to a continuous map from X into the union of Grassmannians of all ranks up to r.
   • The cone map has degree m on the link circle; the umbrella map extends the ruling direction continuously onto the singular line.
6. Nontriviality in Ks:
   • Use the exact sequence for the pair (X, singular set):
     0 → Ks(singular) → Ks(X) → Ks(regular) → 0.
   • For the cone, the extra Z corresponds to holonomy.
   • For the umbrella, the ruling line gives a new extension class.

Next Steps

1. Compute Ks(C(S1)) explicitly.
   Derive the group structure using a clutching argument or Mayer-Vietoris sequence. Expect Z ⊕ Z.
2. Compute Ks for the Whitney umbrella.
   Set up the exact sequence for (X, singular line) and determine whether the ruling direction bundle represents a nontrivial extension class.
3. Generalize.
   Replace S1 with a general compact manifold Y and analyze how holonomy representations affect Ks(C(Y)).
   Extend the Whitney idea to more complex singular surfaces.
4. Explore analytic and physical analogues.
   Connect these bundles to constructible sheaves, intersection cohomology, or index theory on singular manifolds.
5. Open question.
   Classify all rank-1 stratified line bundles on cones C(Y) in terms of fundamental group representations.
   Determine if every such bundle arises from a local system on Y.

Significance

Although the language of stratified vector bundles sounds abstract, the geometry it captures is one of the most concrete and universal structures in applied science.
Whenever a continuous system encounters a boundary, defect, or self-intersection, it leaves the world of smooth manifolds and enters the world of stratified spaces.

Ordinary vector bundles fail there.
Fields that were once smooth may drop rank, twist discontinuously, or carry residual holonomy.
Stratified bundles are what remain — the consistent continuation of structure through singular geometry.

1. Geometry of Continuity Through Defect

The cone example models what happens when a field circulates around a singular axis, like a vortex, screw dislocation, or phase defect.
Its “rank drop at the apex” mirrors the loss of dimensionality at a defect core — the point where the local space collapses, but global continuity survives through monodromy.
This kind of geometry appears in:

• crystal dislocations and Burgers vectors,
• quantum vortices and Aharonov–Bohm flux lines,
• topological optics and phase singularities in wavefronts.

2. Geometry of Self-Contact and Folding

The Whitney umbrella captures how a surface or constraint manifold can intersect itself and still carry a coherent tangent direction along a singular line.
That is the blueprint for:

• caustics and fold catastrophes in optics,
• elastic surfaces with creases or pinch lines,
• control manifolds where different branches of a system meet smoothly in state space.

3. Data, Rank, and Field Continuation

In signal theory, robotics, and numerical geometry, one often needs to continue a field, a basis, or a coordinate frame through a region where rank changes.
A stratified bundle gives the precise topological language for “continuous but not smooth” continuation.
The same mathematics describes:

• sensor fusion near singular configurations,
• neural or data manifolds with collapsing directions,
• low-rank transitions in numerical models.

4. The Broader Goal

Finding and classifying such spaces is the first step toward a taxonomy of singular geometries that still support structure.
Every explicit example makes Ks(X) a usable invariant rather than an abstract one.
It allows topology to measure how much information a field can preserve when its domain ceases to be smooth.

This is the frontier between pure and applied geometry:
not studying where smoothness holds, but where it breaks and yet coheres.
The stratified cone and Whitney umbrella are the first canonical laboratories for that phenomenon.

Closing Note

The goal here is not to classify every stratified bundle but to provide visible, verifiable examples.
These models turn an abstract definition into something geometric, computable, and intuitive.
They establish a concrete foundation for studying Ks(X) and open the way toward a deeper understanding of vector bundles on singular spaces.

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