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!



Local topology of Whitney stratified spaces
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.


