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 (Sieklucki, 1969, Schepin and Shtanko, 1983, Keesling and Wilson, 1985, Akhmetiev, 1996, [Min97] and [ReSk98]). We obtain a criterion for approximability by embeddings of a PL map S¹ --> R².

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 (see the definition in [Min97]) neither contains transversal self-intersections nor is the standard winding of degree n distinct from -1, 0, 1

A corollary of an analogous criterion for PL maps h :I --> R² [Min97] is the deleted product criterion for approximability by embeddings of PL maps h :I --> R². Denote by I^- the set of pairs (x, y), where x, y belong to I and x is distinct from y. Denote by I^{- h} the subset of I^-, formed by the pairs (x, y) such that hx is distinct from hy. Let h^-:I^{- h} --> S¹ be the map given by the formula h^- (x, y)=(hx-hy)/||hx-hy||.

Theorem 2. A PL map h :I --> R² is approximable by embeddings if and only if the map h^-:I^-h --> S¹ extends homotopically to a map I^- --> S¹ (or, equivalently, the van Kampen obstruction v(h)=0; see the definition in [ReSk98]).

References

[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.

[ReSk98] D. Repovs and A. B. Skopenkov, A deleted product criterion for approximability of maps by embeddings, Topol. Appl. 87 (1998) 1-19.

Date received: June 26, 2000