Cellularization for exceptional spherical space forms and the flag manifold of $SL_3(\mathbb{R})$

TL;DR

This paper constructs explicit cellular decompositions for certain spherical space forms and the flag manifold of SL_3(R), providing detailed homological descriptions and equivariant decompositions.

Contribution

It introduces explicit equivariant cellular decompositions for spherical space forms and the flag manifold of SL_3(R), advancing understanding of their topological structures.

Findings

Explicit cellular decomposition of (4n-1)-sphere with binary polyhedral groups

Description of the cellular homology chain complex

Equivariant decomposition of the flag manifold of SL_3(R)

Abstract

We construct an explicit equivariant cellular decomposition of the $(4n-1)$-sphere with respect to binary polyhedral groups, and describe the associated cellular homology chain complex. As a corollary of the binary octahedral case, we deduce an $\mathfrak{S}_3$-equivariant decomposition of the flag manifold of $SL_3(\mathbb{R})$.