Arkadiy Skopenkov

Stability of intersections of graphs in the plane and the van Kampen obstruction

Abstract A map φ : K → R 2 of a graph K is approximable by embeddings , if for each e > 0 there is an e -close to φ embedding f : K → R 2 . Analogous notions were studied in computer science under the names of cluster planarity and weak simplicity . This short survey is intended not only for specialists in the area, but also for mathematicians from other areas. We present criteria for approximability by embeddings (P. Minc, 1997, M. Skopenkov, 2003) and their algorithmic corollaries. We introduce the van Kampen (or Hanani–Tutte) obstruction for approximability by embeddings and discuss its completeness. We discuss analogous problems of moving graphs in the plane apart (cf. S. Spiez and H. Torunczyk, 1991) and finding closest embeddings (H. Edelsbrunner). We present higher dimensional generalizations, including completeness of the van Kampen obstruction and its algorithmic corollary (D. Repovs and A. Skopenkov, 1998).

A characterization of submanifolds by a homogeneity condition

Abstract A very short proof of the following smooth homogeneity theorem of D. Repovs, E.V. Shchepin and the author is presented. Let N be a locally compact subset of a smooth manifold M. Assume that for each two points x , y ∈ N there exists a diffeomorphism h : M → M such that h ( x ) = y and h ( N ) = N . Then N is a smooth submanifold of M.

Codimension Two PL Embeddings of Spheres with Nonstandard Regular Neighborhoods

AbstractFor a given polyhedron K ⊂ M, the notation RM(K) denotes a regular neigh-borhood of K in M. The authors study the following problem: find all pairs (m, k) such that if K is a compact k-polyhedron and M a PL m-manifold, then RM(f(K)) ≅ RM(g(K)) for each two homotopic PL embeddings f, g : K → M. It is proved that RSk+2 (Sk) ≇ Sk × D2 for each k ≥ 2 and some PL sphere Sk ⊂ Sk+2 (even for any PL sphere Sk ⊂ Sk+2 having an isolated non-locally flat point with the singularity Sk-1 ⊂ Sk+1 such that π1(Sk+1 – Sk-1) ≇ ℤ).