Generalized manifolds in products of curves

Abstract

The intent of this article is to distinguish and study some n n -dimensional compacta (such as weak n n -manifolds) with respect to embeddability into products of n n curves. We show that if X X is a locally connected weak n n -manifold lying in a product of n n curves, then rank ⁡ H 1 ( X ) ≥ n \operatorname {rank} H^{1}(X)\ge n . If rank ⁡ H 1 ( X ) = n \operatorname {rank} H^{1}(X)=n , then X X is an n n -torus. Moreover, if rank ⁡ H 1 ( X ) > 2 n \operatorname {rank} H^{1}(X)>2n , then X X can be presented as a product of an m m -torus and a weak ( n − m ) (n-m) -manifold, where m ≥ 2 n − rank ⁡ H 1 ( X ) m\ge 2n-\operatorname {rank} H^{1}(X) . If rank ⁡ H 1 ( X ) > ∞ \operatorname {rank} H^{1}(X)>\infty , then X 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 2 -manifold X X lying in a product of two graphs such that H 2 ( X ) = 0 H^{2}(X)=0 .