Local topology of Whitney stratified spaces

Let $$M$$ be a smooth manifold, let $$\mathcal{P}$$ be a Whitney stratification of $$M$$ and let $$S\subset M$$ be a stratum with closure $$\overline{S}$$.

Question: Does there exist an open neighborhood $$U\subset M$$ of $$\overline{S}$$ such that $$U$$ deformation retracts onto $$\overline{S}$$?

In the case when $$\overline{S}\subset M$$ is a smooth submanifold, this follows from the tubular neighborhood theorem. For another example, let $$M=\mathbb{R}^{2n}$$ and consider the stratification with three strata given by $$S_1=0\in \mathbb{R}^{2n}$$, $$S_2=(\mathbb{R}^n\times\{0\}\cup\{0\}\times\mathbb{R}^n)-S_1$$ and $$S_3=\mathbb{R}^{2n}-\overline{S_2}$$. Then $$M$$ deformation retracts onto $$\overline{S_2}$$ (e.g. along the connected components of the quadics $$||x||^2-||y||^2=t$$ for $$(x,y)\in \mathbb{R}^{2n}$$ and $$t\in\mathbb{R}$$).

I hope that an affirmative answer in general should follow from the Thom/Mather theory of tubular neighborhoods/control data, but I keep getting turned around. A proof or counter-example would be greatly appreciated! Thanks!

Update: For conically stratified manifolds in the sense of Ayala-Francis-Tanaka, this follows from Proposition 8.2.5 of https://arxiv.org/abs/1409.0501. Under the expectation that Whitney stratified spaces should be conically stratified, there should be an analogous result for Whitney stratified manifolds. Is this the case? If so, a reference would be greatly appreciated!

In fact, any Whitney stratified set admits a stratification in the sense of Thom/Mather, cf Mather's notes on topological stability, published in B.A.M.S Volume 49, Number 4, October 2012, Pages 475–506, cf section 8 in particuler p 492 : "Hence it follows from Proposition 7.1 that any Whitney stratified set admits the structure of an abstract stratified set."

Update: The affirmative answer appears as Proposition 3.2 of Quinn, "Homotopically Stratified Spaces" JAMS, 1988.

• Hi David, thanks for the answer. How does admitting a structure of an abstract stratified set imply that for any union of strata there is an open neighborhood of the union which deformation retracts onto it? Mar 19, 2021 at 17:29
• @JesseWolfson : Mather/Thom stratification come by definition with tubular neighborhoods; there is something left to prove to go the closure of a stratum, which should involve taking the argmin of the "distance to strata" functions over all adherent strata to choose to which one project. You might find useful an expository paper of mine (in French, sorry!): perso.math.u-pem.fr/kloeckner.benoit/papiers/… especially definition 3.1 and Theorem 3.2 Mar 19, 2021 at 17:47
• @Benoit, thanks for the reference. The \min distance to strata idea is one I've been trying to implement but getting turned around with. I'll check out your notes! Mar 19, 2021 at 17:52
• Sorry Jesse I was lazy, in fact Thom-Mather stratifications, are part of a larger class of stratified spaces defined by F. Quinn and called "Homotopically stratified spaces" and these stratified spaces have such neighourhoods for a recent treatment : "Strongly stratified homotopy theory" by D. A. Miller can be useful in particular proposition 2.7 (due to F. Quinn). Anyway, as Benoit suggested I think there is a direct proof without using this stuff. Mar 19, 2021 at 17:57
• @David, thanks! I'll check this out! Mar 19, 2021 at 18:03

In a paper written with a collaborator that we have recently uploaded on the arxiv, we show that indeed the conical charts of a Whitney stratified space provided by Thom and Mather induce a conically smooth structure: thus, you may freely apply Proposition 8.2.5 in the paper by Ayala-Francis-Tanaka to get what you want.

• thanks for calling this to my attention! Sep 24, 2021 at 16:50