Stratification

Sketch

We have a function $f:M\to {\mathbb{R}}^n$, with $n$ the number of colours we use. ${\mathbb{R}}^n$ is stratified with $n$ types of strata, according to which coordinate is maximal. We want to look at how $f$ pulls this stratification back to $M$.

Stratified manifolds

Manifold always means smooth manifold.

Definition (pre-stratification)

Let $M$ be a manifold. Let $\mathcal{S}$ be a cover of $M$, consisting of pairwise disjoint smooth submanifolds of $M$, which we call strata. Then we call $\mathcal{S}$ a pre-stratification of $M$.

Definition (Condition b)

Let $M$ a manifold.

Definition (Condition of the frontier)

Let $M$ a manifold and $U$ a subset. We say the frontier of $U$ is the set of limit points of $U$ not contained in $U$, being $\bar{U}-U$. Let ${\mathcal{S}}_M$ be a pre-stratification. We say ${\mathcal{S}}_M$ satisfies the condition of the frontier if for each stratum $X\in {\mathcal{S}}_M$ its frontier is a union of strata.

Definition (locally finiteness)

Let $M$ a manifold and ${\mathcal{S}}_M$ a pre-stratification of $M$. Then we say ${\mathcal{S}}_M$ is locally finite, if each point in $M$ has a neighbourhood which intersects finitely many strata.

Definition (stratified manifold)

Let $M$ a manifold and ${\mathcal{S}}_M$ a pre-stratification of $M$. Then we say that ${\mathcal{S}}_M$ is a stratification of $M$ if:

1. It is locally finite.
2. It satisfies the condition of the frontier.
3. It satisfies condition b.

We then call $(M,{\mathcal{S}}_M)$ a stratified manifold.

Definition (stratified diffeomorphism)

Let $(X,{\mathcal{S}}_X),(Y,{\mathcal{S}}_Y)$ two stratified manifolds. We say a diffeomorphism $f:(X,{\mathcal{S}}_X)\to (Y,{\mathcal{S}}_Y)$ is a stratified diffeomorphism if there exists a bijection $b:{\mathcal{S}}_X\to {\mathcal{S}}_Y$ such that for every $(S,b(S))\in ({\mathcal{S}}_X,{\mathcal{S}}_Y)$, the restriction $f{|}_S:S\to b(S)$ is a diffeomorphism.

Transversality

Definition transversality of vector spaces

Assume We have two vectorsubspaces $A,B\subset X$ of some vector space $X$. We write $A⋔B$ if $A+B=X$.

Definition transversality of manifolds

Assume we have two submanifolds $A,B\subset M$ of some manifold $M$. We write $A⋔B$ if for each $p\in A\cap B,T_{p}A⋔T_{p}B$ (as vector spaces).

Definition 4.1. (p.50) Transversality

Let X, Y be smooth manifolds and $f:X\to Y$ be a smooth mapping. Let $W$ be a submanifold of $Y$ and $x\in X$. Then $f$ intersects $W$ transversally at $x$ (denoted by $f⋔W$ at $x$) if either:

1. $f(x)\notin W$, or
2. $f(x)\in W$, and $T_{f(x)}W⋔(df{)}_x(T_{x}X)$ (as vector spaces).

We write $f⋔W$ on $A$ if $f⋔W$ at $x,\forall x\in A$. We leave the “on …” if $f⋔W$ on the entire $X$.

Definition (Transversal map)

Let $X$ some manifold, and $x\in X$. Let $(Y,{\mathcal{S}}_Y)$ be some stratified manifold. We say that a function $f:X\to Y$ intersects ${\mathcal{S}}_Y$ transversally at $x$ (denoted by $f⋔{\mathcal{S}}_Y$), if for each stratum $S\in {\mathcal{S}}_Y$, $f$ intersects $S$ transversally at $x$.

We write $f⋔{\mathcal{S}}_Y$ on $A$ if $f⋔{\mathcal{S}}_Y$ at $x,\forall x\in A$. We leave the “on …” if $f⋔{\mathcal{S}}_Y$ on the entire $X$.

Claim

Let $X$ a manifold, and $(Y,{\mathcal{S}}_Y)$ a stratified manifold. Then $f⋔{\mathcal{S}}_Y$ is a generic property of $C^{\infty }(X,Y)$.

Stratified model

Definition (stratified model) (Should add notion of “stable model”)

Let $M$ be some stratified manifold. We say that a collection of stratified manifolds $\mathcal{M}$ is a stratified model of $M$, if for every point $p\in M$, there exists a neighbourhood (endowed with the subset stratification¹) for which there exists a stratified diffeomorphism with some component of $\mathcal{M}$.

Definition / Claim (max-stratification) (These have overlap when defined as such so is probably not good)

Let ${\mathcal{S}}_m:=\{U_S|S\subset \{\mathrm{0..}n\}\}$ be a collection of subsets of ${\mathbb{R}}^n$ with $U_S$ given by:

$$U_S:=\{p\in {\mathbb{R}}^n|p_i\in \underset{j}{\max }(p_j),\forall i\in S\}.$$

Then ${\mathcal{S}}_m$ is a stratification, which we will call the max-stratification. Additionally, the dimension of $U_S$ will be given by $\dim (U_S)=n-|S|$.

Footnotes

1. Does a stratification always descent to subsets? ↩