Depth of a point in a manifold with corners - Imported Knowlpedia Content

Definition

Let $M$ be an $n$ -dimensional manifold with corners, and choose a corner chart in which $x\in M$ corresponds to

(u_1,\ldots,u_k,v_{k+1},\ldots,v_n) \in[0,\infty)^k\times\mathbb R^{n-k}.

The depth of $x$ is

\operatorname{depth}_M(x)=\#\{i\in\{1,\ldots,k\}:u_i=0\}.

Compatibility of corner charts makes this number independent of the chosen chart. For $0\leq r\leq n$ , the depth- $r$ stratum is

S^r(M)=\{x\in M:\operatorname{depth}_M(x)=r\}.

Thus depth is intrinsic even though the individual boundary coordinates are not. It measures local codimension in the canonical corner stratification.

Stratification

The decomposition

M=\coprod_{r=0}^n S^r(M)

is a stratification by smooth manifolds without boundary, with $\dim S^r(M)=n-r$ . The depth-zero stratum is the interior. The closure of $S^r(M)$ is the union of strata of depth at least $r$ . These properties are formulated for ordinary corners in Joyce, Definition 2.6.

Examples and products

For the square $[0,1]^2$ , interior points have depth zero, points in open edges have depth one, and vertices have depth two. On a manifold with boundary, only depths zero and one occur. In a product,

\operatorname{depth}_{M\times N}(x,y) =\operatorname{depth}_M(x)+\operatorname{depth}_N(y),

because the vanishing coordinates from the two factors concatenate.

Depth versus a chosen face

References

Dominic Joyce, “A Generalization of Manifolds with Corners,” Advances in Mathematics 299 (2016), 760–862. DOI record. Relevant: §2, Definition 2.6 for depth and depth strata of ordinary manifolds with corners, and §3.4 for the generalized construction.
Dominic Joyce, “On Manifolds with Corners,” final preprint version, 2010. arXiv record. Relevant: §2, boundaries and corners.