On the topology of symmetry sets of smooth submanifolds in $\mathbb{R}^k$

Abstract

We study topology of symmetry sets, conflict sets and medial axes in the case when they have only stable singularities of corank 1. Singularities of these sets satisfy various conditions of coexistence. For example, isolated singularities and singularities forming smooth non-closed curves define a graph. If this graph is finite, then there is the following incidence relation: the sum of the local degrees of vertices of the graph is equal to the doubled number of its edges (the local degree of a vertex is the number of edges that are incident to this vertex; loops are counted twice). We give many-dimensional generalizations of this relation for sets mentioned above. These generalizations follow from some general facts on coexistence of wave front singularities found recently by the author.