# How to understand and use the Whitney stratification theorem?

Why can the Whitney stratification theorem can show that the union of submanifolds of co-dimension at least 1 (as shown in the figure)?

Whitney stratification theorem: Any semialgebraic set $$V\subseteq\mathbb R^n$$ admits a canonical Whitney stratification $$\mathfrak X$$ having finitely many semialgebraic strata.

And could you give some examples on how to understand and use the Whitney stratification theorem?

Proof. Let us start with $$f\in C^q$$ with $$q\ge3$$. First, critical points of a function $$f$$ are such that $$D_if(x)=0,i=1,2$$. Furthermore, the $$J^2(\mathbb R^m,\mathbb R)$$ bundle associated to $$f$$ is diffeomorphic to $$\mathbb R^m×\mathbb R×\mathbb R^m×\mathbb R^{m(m+1)/2}$$ and the $$2$$-jet extension of $$f$$ at any point $$x\in\mathbb R^m$$ is given by $$(x,f(x),Df(x),D^2f(x))$$.

Now, let us denote by $$S(k)$$ the space of $$k×k$$ symmetric matrices, and consider the subset of $$J^2(\mathbb R^m,\mathbb R)$$ defined by $$\mathcal D_i=\mathbb R^m×\mathbb R×\{0\}×Z_i(m_i)×\mathbb R^{m_1×m_2}×S(m-m_i)$$ where $$Z_i(m_i)=\{A\in S(m_i)|\det(A)=0\}$$. The set $$Z(m_i)$$ is algebraic; hence, we can use the Whitney stratification theorem (Gibson et al., 1976, Ch. 1, Thm. 2.7) to get that each $$Z(m_i)$$ is the union of submanifolds of co-dimension at least $$1$$. Hence, it is the union of sub-manifolds of codimension at least one and, in turn, $$\mathcal D_i$$ is the union of sub-manifolds of codimension at least $$m+1$$. Thus, it follows from the Jet Transversality Theorem C.1 (by way of proposition C.2 since $$m+1>m$$) that for a generic function $$f$$, the image of the $$2$$-jet extension $$j^2f$$ is disjoint from $$\mathcal D_i$$. Hence, for such an $$f$$, for each $$x$$ that is a critical point, the Hessian of $$f$$ is such that $$\det(D_i^2f(x))\ne0$$.

• That was a mouthful for me to type; it would have been better if you typed all that out while writing your question in the first instance. May 7, 2021 at 5:21
• @ParclyTaxel I am very sorry about that. It is the first time I ask the question. Thank you so much for helping me to modify the question!
– YRS
May 7, 2021 at 5:34