Examples of rational toral rank complex

In "A Hosse diagram for rational toral tanks," we see a CW complex ${\mathcal T}(X)$, which gives a rational homotopical classification of almost free toral actions on spaces in the rational homotopy type of $X$ associated with rational toral ranks and also presents certain relations in them. We call it the {\it rational toral rank complex} of $X$. It represents a variety of toral actions. In this note, we will give effective 2-dimensional examples of it when $X$ is a finite product of odd spheres. This is a combinatorial approach in rational homotopy theory.Comment: 8 page

A Lower Bound for the Rational LS-category of a Coformal Elliptic Space

We give a lower bound for the rational LS-category of certain spaces, including the coformal elliptic ones, in terms of the dimension of its total rational cohomology.</p

Rational toral rank of a map

Let $X$ and $Y$ be simply connected CW complexes with finite rational cohomologies. The rational toral rank $r_0(X)$ of a space $X$ is the largest integer $r$ such that the torus $T^r$ can act continuously on a CW-complex in the rational homotopy type of $X$ with all its isotropy subgroups finite \cite{H}. As a rational homotopical condition to be a toral map preserving almost free toral actions for a map $f:X\to Y$, we define the rational toral rank $r_0(f)$ of $f$, which is a natural invariant with $r_0(id_X)=r_0(X)$ for the identity map $id_X$ of $X$. We will see some properties of it by Sullivan models, which is a free commutative differential graded algebra over \Q \cite{FHT}.Comment: 8 page

A Lower Bound for the LS Category of a Formal Elliptic Space

We give a lower bound for the LS category of a formal elliptic space in terms of its rational cohomology.</p