Cellular homology of compact groups: Split real forms

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 $G_2$.