Thom–Mather stratified space

A Thom–Mather stratified space is a triple ${\displaystyle (V,{\mathcal {S}},{\mathfrak {J}})}$ where ${\displaystyle V}$ is a topological space (often we require that it is locally compact, Hausdorff, and second countable), ${\displaystyle {\mathcal {S}}}$ is a decomposition of ${\displaystyle V}$ into strata,

${\displaystyle V=\bigsqcup _{X\in {\mathcal {S}}}X,}$

and ${\displaystyle {\mathfrak {J}}}$ is the set of control data ${\displaystyle \{(T_{X}),(\pi _{X}),(\rho _{X})\ |X\in S\}}$ where ${\displaystyle T_{X}}$ is an open neighborhood of the stratum ${\displaystyle X}$ (called the tubular neighborhood), ${\displaystyle \pi _{X}:T_{X}\to X}$ is a continuous retraction, and ${\displaystyle \rho _{X}:T_{X}\to [0,+\infty )}$ is a continuous function. These data need to satisfy the following conditions.

1. Each stratum ${\displaystyle X}$ is a locally closed subset and the decomposition ${\displaystyle S}$ is locally finite.
2. The decomposition ${\displaystyle S}$ satisfies the axiom of the frontier: if ${\displaystyle X,Y\in {\mathcal {S}}}$ and ${\displaystyle Y\cap {\overline {X}}\neq \emptyset }$, then ${\displaystyle Y\subseteq {\overline {X}}}$. This condition implies that there is a partial order among strata: ${\displaystyle Y<X}$ if and only if ${\displaystyle Y\subset {\overline {X}}}$ and ${\displaystyle Y\neq X}$.
3. Each stratum ${\displaystyle X}$ is a smooth manifold.
4. ${\displaystyle X=\{v\in T_{X}\ |\ \rho _{X}(v)=0\}}$. So ${\displaystyle \rho _{X}}$ can be viewed as the distance function from the stratum ${\displaystyle X}$.
5. For each pair of strata ${\displaystyle Y<X}$, the restriction ${\displaystyle (\pi _{Y},\rho _{Y}):T_{Y}\cap X\to Y\times (0,+\infty )}$ is a submersion.
6. For each pair of strata ${\displaystyle Y<X}$, there holds ${\displaystyle \pi _{Y}\circ \pi _{X}=\pi _{Y}}$ and ${\displaystyle \rho _{Y}\circ \pi _{X}=\rho _{Y}}$ (both over the common domain of both sides of the equation).

One of the original motivations for stratified spaces were decomposing singular spaces into smooth chunks. For example, given a singular variety ${\displaystyle X}$, there is a naturally defined subvariety, ${\displaystyle \mathrm {Sing} (X)}$, which is the singular locus. This may not be a smooth variety, so taking the iterated singularity locus ${\displaystyle \mathrm {Sing} (\mathrm {Sing} (X))}$ will eventually give a natural stratification. A simple algebreo-geometric example is the singular hypersurface

     ${\displaystyle {\text{Spec}}\left(\mathbb {C} [x,y,z]/\left(x^{4}+y^{4}+z^{4}\right)\right)\xleftarrow {(0,0,0)} {\text{Spec}}(\mathbb {C} )}$

where ${\displaystyle {\text{Spec}}(-)}$ is the prime spectrum.

• Singularity theory
• Whitney conditions
• Stratifold
• Intersection homology
• Thom's first isotopy lemma
• stratified space

• Goresky, Mark; MacPherson, Robert Stratified Morse theory, Springer-Verlag, Berlin, 1988.
• Goresky, Mark; MacPherson, Robert Intersection homology II, Invent. Math. 72 (1983), no. 1, 77--129.
• Mather, J. Notes on topological stability, Harvard University, 1970.
• Thom, R. Ensembles et morphismes stratifiés, Bulletin of the American Mathematical Society 75 (1969), pp.240-284.
• Weinberger, Shmuel (1994). The topological classification of stratified spaces. Chicago Lectures in Mathematics. Chicago, IL: University of Chicago Press. ISBN 9780226885667.