Gluing of multiple Alexandrov spaces

Abstract

In this paper we discuss the sufficient and necessary conditions for multiple Alexandrov spaces being glued to an Alexandrov space. We propose a Gluing Conjecture, which says that the finite gluing of Alexandrov spaces is an Alexandrov space, if and only if the gluing is by path isometry along the boundaries and the tangent cones are glued to Alexandrov spaces. This generalizes Petrunin's Gluing Theorem. Under the assumptions of the Gluing Conjecture, we classify the 2-point gluing over (n − 1, ǫ)-regular points as local separable gluing and the gluing near un-glued (n − 1, ǫ)-regular points as local involutional gluing. We also prove that the Gluing Conjecture is true if the complement of (n − 1, ǫ)-regular points is discrete in the glued boundary. In particular, this implies the general Gluing Conjecture as well as a new Gluing Theorem in dimension 2.