The intent of this article is to distinguish and study some ndimensional compacta (such as weak n-manifolds) with respect to embeddability into products of n curves. We show that if X is a locally connected weak n-manifold lying in a product of n curves, then rank H1(X) ≥ n. If rankH1(X) = n, then X is an n-torus. Moreover, if rank H1(X) < 2n, then X can be presented as a product of an m-torus and a weak (n-m)-manifold, where m ≥ 2n - rank H1(X). If rank H1(X) < ∞, then X is a polyhedron. It follows that certain 2-dimensional compact contractible polyhedra are not embeddable in products of two curves. On the other hand, we show that any collapsible 2-dimensional polyhedron embeds in a product of two trees. We answer a question of Cauty proving that closed surfaces embeddable in a product of two curves embed in a product of two graphs. We construct a 2-dimensional polyhedron that embeds in a product of two curves but does not embed in a product of two graphs. This solves in the negative another problem of Cauty. We also construct a weak 2-manifold X lying in a product of two graphs such that H2(X) = 0.
ASJC Scopus subject areas
- Applied Mathematics