Boundary face of a manifold with corners - Imported Knowlpedia Content

Definition

Let $X$ be a manifold with corners. At $x\in X$ , a local boundary component is the germ at $x$ of one of the coordinate hypersurfaces $u_i=0$ in a corner chart for which $u_i(x)=0$ . The abstract boundary $\partial X$ consists of pairs $(x,\beta)$ , where $\beta$ is such a local component. A boundary face is a connected component of $\partial X$ ; its image in $X$ is the closure of the corresponding connected depth-one piece. This definition retains which hypersurface is chosen when several meet at a corner.

Incidence at corners

If $x$ has depth $r$ , exactly $r$ local boundary components lie over it. Thus the natural map $\partial X\to X$ need not be injective: a corner point appears once for each incident face. Iterating the abstract-boundary construction records ordered choices of several incident hypersurfaces and refines the corner stratification Joyce, §2, Definitions 2.5–2.8.

Examples

For a square, the four closed edges are boundary faces, and each vertex has two preimages in the abstract boundary, one for each incident edge. For a disk, the boundary is one face. Merely taking the set-theoretic subset $\{x:\operatorname{depth}(x)>0\}$ loses the multiplicity at a square’s vertices and therefore does not encode the same incidence data.

Conventions and scope

References

Dominic Joyce, “On Manifolds with Corners,” in Advances in Geometric Analysis, Advanced Lectures in Mathematics 21, International Press, 2012. Author preprint. Relevant: §2, Definitions 2.5–2.8 and Remark 2.11, local boundary components, abstract boundaries, and corner spaces.
Richard B. Melrose, Differential Analysis on Manifolds with Corners, unfinished book manuscript. Author-hosted manuscript. Relevant: Chapter 1, boundary hypersurfaces and manifolds with faces.