I just started learning about these things, so there is a chance I might have misunderstood some things. My apologies if that is the case.
Some context. Suppose that we are given a differentiable map $f:X\to Y$. For $r\geq 0$ one can define $S_{r}(f)$ as the set of points $x$ in which $rk(Df(x))=\min(dim(X),dim(Y))-r$. By checking that suitable subsets of the first order jet space $J^{1}(X,Y)$ are sub-manifolds one can use Thom's transversality theorem to show that for generic $f$ (that is, for any $f$ in some comeager set in the Whitney topology) the sets $S_{r}(f)$ are sub-manifolds of $X$ (with $S_{0}(f)$ and $S_{r}$ of positive codimension for $r>0$). For any such $f$ one can go a step further and given any $r>0$ for which $S_{r}\neq\emptyset$ and any $s\geq 0$ consider the set $S_{r,s}$ of points $x\in S_{r}$ in which the rank of the map $$Df(x)_{\restriction T_{x}S_{r}(f)}:T_{x}S_{r}(f)\to T_{f(x)}Y$$ differs by $s$ from the maximum possible rank allowed by the dimension of its domain and codomain. It turns out that for $f$ in a generic subset $S_{r,s}$, $s>0$ are once again sub-manifolds of $S_{r}$. It is a result by Boardman that generically this process can be continued as long as dimension allows for it, resulting in a stratification of $X$ by sets of the form $S_{I}(f)$, where $I=(i_0,\dots i_{ḱ-1},0)$ is some multi-index with $i_{k}>0$. We will refer to any $f$ satisfying all the conditions as a Boardman map.
Question. My question regards the geometric shape of individual strata. I am interested in understanding whether the following property holds.
- Let $f:X\to Y$ be a Boardman map with compact domain and $S:=S_{I}(f)$ one of the strata. Then there exists some compact manifold with boundary $M$ such that for any $\epsilon>0$ there is some diffeomorphic copy $M_{\epsilon}$ of $M$ with $$S\setminus N_{\epsilon}(cl(S)\setminus S)\subset M_{\epsilon}\subset S$$ where $N_{\epsilon}(A)=\{x\,|\,d(x,A)<\epsilon\}$. (As a matter of fact, any compact $M$ whose interior is a manifold and which naturally lives in the interior of a copy of itself would suffice.)
The (potentially very wrong) intuition behind this is that any such $S$ should be diffeomorphic to the interior of a generalized manifold with corners of some sort (modeled by a general "polytope with cusps" around the boundary, whatever that means) which then surjects onto $cl(S)$ in a way compatible with the embedding $S\hookrightarrow cl(S)$.
How wrong is this intuition? What is the most informative thing one can say in general about the way each stratum sits in its closure?
