On homological properties of locally polar spaces

Recall that the flag complex of a geometry is the complex whose points are objects and simplices are flags of this geometry. A geometry is Cohen–Macaulay if the reduced homology of its flag complex vanishes in all dimensions except for the top one, and all residues also have this property. It is proved in the article that the locally polar spaces of order two are Cohen–Macaulay. Results of this kind have applications to studying cohomology of groups acting on geometries.