A Criterion for Approximability by Embeddings of Cycles in the Plane
by
Mikhail Skopenkov
Moscow State University

The problem of approximability by embeddings appeared in studies of embeddability of compacta into R² (K. Sieklucki, 1969, E. V. Schepin and M. A. Shtanko, 1983, D. Repovs and A. B. Skopenkov, 1996).

Theorem 1. Let h :S¹ --> R² be a PL map, which is simplicial for some triangulation of S¹ with m vertices. The map h is approximable by embeddings if and only if for each k=0, ..., m⁴ the k-th derivative d^k( h ):S¹ --> R² neither contains transversal self-intersections nor is the standard winding of degree n distinct from -1, 0, 1 (for h :I --> R² see [Min97]).

Let us introduce definitions. By h :K --> G we denote a simplicial map of a graph K onto a graph G. We assume that G is embedded into a thickening of G (i.e., N is a regular neighbourhood of the graph G). The derivative graph D(G) of a graph G is defined as follows. Each of its vertices e^* corresponds to an edge e of G. Vertices e^* and f^* are joined by an edge in D(G), if edges e and f are adjacent. For the thickening N of G is evidently defined the derivative thickening D(N) (for accurate definition see [Min97]). Define the graph D( h , K) . Each of its vertices a^* corresponds to some connected component a of h ^-1e such that h a is an edge e of G. Vertices a^* and b^* are joined by an edge, if and only if a and b intersect (as the subgraphs of G). Define the map d( h ): D( h , K) --> D(G) on the vertices of D( h , K) by setting d( h ) (a^*)=( h a)^*, and then extend d( h ) linearly over the edges of D( h , K). Define d^k( h ) inductively by the formula d^k( h )=d(d^k-1( h )).

Theorem 2. A simplicial map h :I --> R² is approximable by embeddings if and only if the van Kampen obstruction v(h)=0 (see the definition in [CRS98]).

[CRS98] A. Cavicchioli, D. Repovs and A. B. Skopenkov, Open problems on graphs, arising from geometrical topology Topol. Appl. 84 (1998), 207-226.

[Min97] P. Minc, Embedding simplicial arcs into the plane, Topol. Proc. 22 (1997), 305-340.

Date received: June 19, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caeu-29.