Problems in embedding theory of 2-polyhedra
by
Krešmir Horvatič
Department of Mathematics, University of Zagreb

An embedding of a (compact) polyhedron is a PL injection of it into a given ambient space. Every n-polyhedron embeds into the Euclidean space E²ⁿ⁺¹, and in general the result is the best possible. Shapiro and Wu constructed an obstruction Ob(P) for the embedability in spaces of lower dimensions and showed, that for n not equal to 2, an n-polyhedron P embeds into E²ⁿ if and only if Ob(P) vanishes. The condition is hard to verify in concrete situations.

Instead, a lot of sufficient conditions were found. Among them, there are two, which are independent. If n not equal to 2, Hⁿ(P-Int A)=0 (where A is an n-simplex in a triangulation of P; Horvatiæ) or Hⁿ(P)=cyclic (Rees; Wilson) imply embedability of n-polyhedron into E²ⁿ. The theorems cover all other known results, as special cases.

The dimension n=2 is an exception, because this case is 2n=n+2, i.e. the double dimension coincides with codimension 2, in which appears nonstandard phenomena in geometric topology (e.g. knots). Freedman, Kruchal and Teichner (1994) found polyhedron P (with 14 vertices!) showing, that Ob(P)=0 is not a sufficient condition in the Shapiro-Wu theorem, when n=2. On the other hand, Kranjc saved the Reed-Wilson theorem, in this dimension, too. So, natural questions arise:

Problem 1: Is H²(P)=cyclic group also necessary for the embedability of a 2-polyhedron P into E⁴?
Problem 2: Is H²(P-Int A)=0 a necessary and sufficient condition for a 2-polyhedron P, to be embeddable into E⁴?

Date received: March 21, 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 # caex-44.