Generalized manifolds in products of curves

TL;DR

This paper investigates the structure and embeddability of special n-dimensional continua in products of curves, establishing bounds on cohomology ranks, and providing examples and counterexamples related to embeddings in products of graphs.

Contribution

It characterizes when certain n-manifolds can be represented as products of lower-dimensional spaces and answers open questions about embeddings in products of curves and graphs.

Findings

Rank of H^1(X) is at least n for locally connected quasi n-manifolds.

Certain 2D contractible polyhedra cannot embed in products of two curves.

Any collapsible 2D polyhedron embeds in a product of two trees.

Abstract

The intent of this article is to study some special $n$-dimensional continua lying in products of $n$ curves. (The paper is an improved version of a portion of \cite{K-K-S}.) We show that if $X$ is a locally connected, so-called, quasi $n$-manifold lying in a product of $n$ curves then rank of $H^1(X)\ge n$. Moreover, if $\rank H^1(X)<2n$ then $X$ can be represented as a product of an $m$-torus and a quasi $(n-m)$-manifold, where $m\ge2n-\rank H^1(X)$. It follows that certain 2-dimensional contractible polyhedra are not embeddable in products of two curves. On the other hand, we show that any collapsible 2-dimensional polyhedron can be embedded in a product of two trees. We answer a question of R. Cauty proving that closed surfaces embeddable in products of two curves can be also embedded in products of two graphs. On the other hand, we construct an example of a 2-dimensional polyhedron which can be embedded in a product of two curves though it is not embeddable in any product of two graphes. This solves in the negative another problem of Cauty.