A Hasse diagram for rational toral ranks

TL;DR

This paper introduces a Hasse diagram structure for classifying rational toral ranks of a simply connected CW complex, providing a new combinatorial perspective on almost free toral actions and their organization.

Contribution

It constructs a Hasse diagram for the poset of rationalized orbit spaces, linking algebraic invariants with topological structures in a novel way.

Findings

Defines a partial order on rationalized orbit spaces based on Borel fibrations.

Constructs a graph from the Hasse diagram and explores its properties.

Proposes viewing the graph as the skeleton of a CW complex.

Abstract

Let $X$ be a simply connected CW complex with finite rational cohomology. For the finite quotient set of rationalized orbit spaces of $X$ obtained by almost free toral actions, ${\mathcal T}_0(X)=\{[Y_i] \}$, induced by an equivalence relation based on rational toral ranks, we order as $[Y_i]<[Y_j]$ if there is a rationalized Borel fibration $Y_i\to Y_j\to BT^n_{\Q}$ for some $n>0$. It presents a variation of almost free toral actions on $X$. We consider about the Hasse diagram ${\mathcal H}(X)$ of the poset ${\mathcal T}_0(X)$, which makes a based graph $G{\mathcal H}(X)$, with some examples. Finally we will try to regard $G{\mathcal H}(X)$ as the 1-skeleton of a finite CW complex ${\mathcal T}(X)$ with base point $X_{\Q}$.