Corner stratification - Imported Knowlpedia Content

Definition

Let $M$ be an $n$ -dimensional manifold with corners. Its corner stratification is the partition

M=\coprod_{r=0}^n S^r(M),\qquad S^r(M)=\{x\in M:\operatorname{depth}_M(x)=r\},

where the depth of $x$ is the number of vanishing boundary coordinates in any corner chart at $x$ . Each $S^r(M)$ , with its induced smooth structure, is an $(n-r)$ -dimensional smooth manifold without boundary, possibly empty or disconnected. The depth-zero stratum is the interior; positive-depth strata record the open pieces where a fixed number of local boundary hypersurfaces meet.

Local model and incidence

In the orthant $[0,\infty)^k\times\mathbb R^{n-k}$ , the depth- $r$ stratum is the union of loci obtained by setting exactly $r$ of the first $k$ coordinates to zero and keeping the others positive. A limit of depth- $r$ points can have depth greater than $r$ , so

\overline{S^r(M)}\subseteq\bigcup_{s\geq r}S^s(M).

Equality need not hold globally when components or faces do not meet.

Examples and products

For a square, the interior, open edges, and vertices are respectively the depth-zero, depth-one, and depth-two strata. For manifolds with corners $M$ and $N$ , depths add on the product manifold, so the depth- $r$ stratum of $M\times N$ is the disjoint union of $S^i(M)\times S^j(N)$ over $i+j=r$ .

Strata versus faces

References

Dominic Joyce, “On Manifolds with Corners,” final preprint version, 2010. arXiv record. Relevant: §§2–3, depth, strata, boundaries, and corner functors.
Dominic Joyce, “A Generalization of Manifolds with Corners,” Advances in Mathematics 299 (2016), 760–862. DOI record. Relevant: §2, ordinary corner stratifications used as the comparison case.