Cellular homology of compact groups: Split real forms

TL;DR

This paper computes the cellular homology of maximal compact subgroups of split real form Lie groups using Bruhat and Schubert cells, providing explicit formulas and applications to SO(3) and SO(4).

Contribution

It introduces algebraic formulas for boundary maps and characterizes incidence relations in the cellular homology of these groups, extending previous work on flag manifolds.

Findings

Calculated cellular homology of SO(3) and SO(4).

Extended formulas for boundary maps to maximal compact subgroups.

Characterized incidence order between Schubert cells.

Abstract

In this article, we use the Bruhat and Schubert cells to calculate the cellular homology of the maximal compact subgroup $K$ of a connected semisimple Lie group $G$ whose Lie algebra is a split real form. We lift to the maximal compact subgroup the previously known attaching maps for the maximal flag manifold and use it to characterize algebraically the incidence order between Schubert cells. We also present algebraic formulas to compute the boundary maps which extend to the maximal compact subgroups similar formulas obtained in the case of the maximal flag manifolds. Finally, we apply our results to calculate the cellular homology of $\mbox{SO}(3)$ as the maximal compact subgroup of $\mbox{SL}(3, \mathbb{R})$ and the cellular homology of $\mbox{SO}(4)$ as the maximal compact subgroup of the split real form $G_2$.