Examples of rational toral rank complex

TL;DR

This paper provides explicit 2-dimensional examples of the rational toral rank complex for finite products of odd spheres, illustrating a combinatorial approach to classifying toral actions in rational homotopy theory.

Contribution

It introduces concrete examples of the rational toral rank complex for specific spaces, enhancing understanding of its structure and applications.

Findings

Explicit 2-dimensional examples of the rational toral rank complex.

Demonstration of a combinatorial approach in rational homotopy theory.

Insights into toral actions on products of odd spheres.

Abstract

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.