Stratified Space


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Typical examples of stratified spaces include singular solutions to polynomial or real-analytic equations.

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The application of Whitney's Theorem to signal localization is questionable in practice. Signals do not propagate unendingly, and the physical realities of signal reflection/echo, multi-bounce, and diffraction conspire to make manifold theory suboptimal in this setting. The addition of signal noise further frustrates a differential-topological approach.

Assumption that it network domain $\mathcal{D} = \{D_i\}_i$ is a manifold is a poor assumption. In realistic settings, the domain has a boundary: signals are bouncing off of
walls, building exteriors, and other structures that, at best, are piecewise-manifolds.

A setting for stratified space is a smooth n-manifold with boundary is a space locally homeomorphic to either $\mathbb{R}^n$ or $\mathbb{R}^{n-1} \times [0,\infty)$, with the usual compatibility constraints required for a smooth structure. The boundary $\partial \mathcal{D}$ is therefore a manifold of dimension $n-1$. Further generalizing, an n-manifold with corners is a space, each point of which has a neighborhood locally homeomorphic to:
$$\{ x \in \mathbb{R}^n\; :\; x_i \geq 0,\; i = 1, \cdots, m\}$$
for some $0 \leq m \leq n$, where $m$ may vary point to point. The analogues of smoothings, derivatives, tangent and cotangent bundles are straightforward to generate.

The boundary $\partial M$ of a manifold with corners $M$ no longer has the structure of a smooth manifold, as, e.g., is clear in the case of a cube or other platonic solid. Note however, that such a boundary is assembled from manifolds of various dimensions, suitably glued together. Such piecewise-manifolds are common in applications.

Consider the solution to a polynomial equation $p(x) = 0$, for $x \in \mathbb{R}^n$. An application of transversality theory shows that the solution set is, for generic choices of coefficients of $p$, a manifold with dimension $n-1$. however, nature does not always deal out such conveniences. Many applications call for the solution to a specific polynomial equation. The null set of a polynomial, even when not a true manifold, can nevertheless be decomposed into manifolds of various dimensions, glued together in a particular manner.

There is a hierarchy of such stratified spaces which deviate from the smooth regularity of a manifold. An intuitive definition of a stratified space is a space $X$, along with finite partition $X = \cup_i X_i$, such that each $X_i$ is a manifold.

More physical examples are readily generated. Recall the setting of planar linkage configuration spaces.


Topics in Topological Data Analysis

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