A deleted product criterion for approximability of maps by embeddings
by
A. Skopenkov
Moscow State University, Russia
Coauthors: D. Repovs

We prove the following theorem: Suppose that m >= 3(n+1)/2 and that f : K --> R^m is a PL map of an n-dimensional finite polyhedron in R^m. Then f is approximable by embeddings if and only if there exists an equivariant homotopical extension

\Phi: ~ K   --> Sm-1

of the map

~ f   : ~ K   f   --> Sm-1,

defined by

~ f   (x, y) = (fx-fy) / |fx-fy|,

where

~ K   f   = { (x, y) in K ×K | fx =/= fy }.

Our result is a controlled version of the classical deleted product criterion for embeddability of polyhedra in R^m. It also is a generalization of E.V. Schepin and the authors' results on uncountable collections of continua and their span. We also construct the van Kampen obstruction for approximability of a PL map by embeddings.

Date received: May 20, 1997

Copyright © 1997 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 # caao-09.