A Hasse diagram for rational toral ranks

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 $X_{\Q}$.

15 pages