
Typical examples of stratified spaces include singular solutions to polynomial or realanalytic equations.
Motivation
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, multibounce, and diffraction conspire to make manifold theory suboptimal in this setting. The addition of signal noise further frustrates a differentialtopological 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 piecewisemanifolds.
A setting for stratified space is a smooth nmanifold with boundary is a space locally homeomorphic to either $\mathbb{R}^n$ or $\mathbb{R}^{n1} \times [0,\infty)$, with the usual compatibility constraints required for a smooth structure. The boundary $\partial \mathcal{D}$ is therefore a manifold of dimension $n1$. Further generalizing, an nmanifold 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 piecewisemanifolds 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 $n1$. 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.
