Dimension of graphoids of rational vector-functions

TL;DR

This paper investigates the topological and cohomological dimensions of graphoids generated by families of rational functions, revealing their complex structure and providing examples of spaces with specific dimensional properties.

Contribution

It establishes the topological dimension of graphoids for rational functions as 2 and shows that certain families yield graphoids with cohomological dimension 1, illustrating their nuanced dimensional characteristics.

Findings

Graphoids have topological dimension 2.

Certain families produce graphoids with cohomological dimension 1.

These spaces are not dimensionally full-valued, similar to Pontryagin surfaces.

Abstract

Let $F$ be a countable family of rational functions of two variables with real coefficients. Each rational function $f\in F$ can be thought as a continuous function $f:dom(f)\to\bar R$ taking values in the projective line $\bar R=R\cup\{\infty\}$ and defined on a cofinite subset $dom(f)$ of the torus $\bar R^2$. Then the family $\F$ determines a continuous vector-function $F:dom(F)\to\bar R^F$ defined on the dense $G_\delta$-set $dom(F)=\bigcap_{f\in F}dom(F)$ of $\bar R^2$. The closure $\bar\Gamma(F)$ of its graph $\Gamma(F)=\{(x,f(x)):x\in dom(F)\}$ in $\bar R^2\times\bar R^F$ is called the {\em graphoid} of the family $F$. We prove the graphoid $\bar\Gamma(F)$ has topological dimension $dim(\bar\Gamma(F))=2$. If the family $F$ contains all linear fractional transformations $f(x,y)=\frac{x-a}{y-b}$ for $(a,b)\in Q^2$, then the graphoid $\bar\Gamma(F)$ has cohomological dimension $dim_G(\bar\Gamma(F))=1$ for any non-trivial 2-divisible abelian group $G$. Hence the space $\bar\Gamma(F)$ is a natural example of a compact space that is not dimensionally full-valued and by this property resembles the famous Pontryagin surface.