A category of manifolds that includes Polygonal domains - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T17:32:34Z http://mathoverflow.net/feeds/question/103785 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/103785/a-category-of-manifolds-that-includes-polygonal-domains A category of manifolds that includes Polygonal domains Martin 2012-08-02T14:17:38Z 2012-08-02T17:18:12Z <p>The prime motivation to introduce the category of manifolds with corners is to have a convenient theory for the analysis on simplices that is as powerful as for smooth manifolds (with boundaries).</p> <p>As far as I understand, polygonally bounded subdomains of \$\mathbb R^n\$ are <strong>not</strong> submanifolds with corners in general, at least for two reasons: (i) The category of manifolds with corners does not include "inward corners" (ii) The number of polygonal boundary pieces that meet at a common point can be arbitrarly large.</p> <p>Is there a category of manifolds which contains arbitrary polygonal domains (where the boundary pieces may be curved)?</p> http://mathoverflow.net/questions/103785/a-category-of-manifolds-that-includes-polygonal-domains/103791#103791 Answer by Rafe Mazzeo for A category of manifolds that includes Polygonal domains Rafe Mazzeo 2012-08-02T14:59:21Z 2012-08-02T16:21:22Z <p>There is a class of what I call "smoothly stratified spaces". This is a bit less general than the larger class of stratified spaces (satisfying Thom-Mather axioms) in that it doesn't allow cusps. These spaces come up in many settings, but definitely include all polygonal and polyhedral domains. They are the setting for analysis in Cheeger's old paper `Spectral geometry of cone-like spaces' JDG 1983 (?), and are also discussed in detail in a recent paper of mine with Albin, Leichtnam and Piazza (The signature package on Witt spaces, just came out in Ann ENS). One point is that if you have a polyhedron, e.g. in \$\mathbb{R}^3\$, with vertices which are more than trivalent, then it is simply not a manifold with corners, but you can think of the local structure near each vertex as a cone over a spherical polygon. </p>