  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 $(n-1,\epsilon)$-regular
points as local separable gluing and the gluing near un-glued
$(n-1,\epsilon)$-regular points as local involutional gluing. We also prove
that the Gluing Conjecture is true if the complement of
$(n-1,\epsilon)$-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.

Differential Geometry;Metric Geometry