Generalized manifolds in products of curves

Abstract:

The intent of this article is to distinguish and study some $n$-dimensional 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 $\operatorname {rank} H^{1}(X)\ge n$. If $\operatorname {rank} H^{1}(X)=n$, then $X$ is an $n$-torus. Moreover, if $\operatorname {rank} H^{1}(X)<2n$, then $X$ can be presented as a product of an $m$-torus and a weak $(n-m)$-manifold, where $m\ge 2n-\operatorname {rank} H^{1}(X)$. If $\operatorname {rank} H^{1}(X)<\infty$, 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 $H^{2}(X)=0$.