Examples of Rational Toral Rank Complex

Toshihiro Yamaguchi

doi:10.1155/2012/867247

Abstract

There is a CW complex 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 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.

Highlights

Let X be a connected CW complex with dim H∗ X; Q < ∞ and r0 X be the rational toral rank of X, which is the largest integer r such that an r-torus T r S1 × · · · × S1 r-factors can act continuously on a CW-complex Y in the rational homotopy type of X with all its isotropy subgroups finite such an action is called almost free 1
Can hold for a formal space X and an integer n > 1 2. It must appear as one phenomenon in a variety of almost free toral actions
The set of 0-cells T0 X is the finite sets { s, r ∈ Z≥0 × Z≥0} where the point Ps,r of the coordinate s, r exists if there is a model ΛW, dW ∈ Xr and r0 ΛW, dW r0 X − s − r

Summary

Introduction

Let X be a connected CW complex with dim H∗ X; Q < ∞ and r0 X be the rational toral rank of X, which is the largest integer r such that an r-torus T r S1 × · · · × S1 r-factors can act continuously on a CW-complex Y in the rational homotopy type of X with all its isotropy subgroups finite such an action is called almost free 1. The set of 0-cells T0 X is the finite sets { s, r ∈ Z≥0 × Z≥0} where the point Ps,r of the coordinate s, r exists if there is a model ΛW, dW ∈ Xr and r0 ΛW, dW r0 X − s − r. The 2 cell is given if there is a homotopy commutative diagram of restrictions ( ΛW, dW )

Results

Conclusion