Homotopy properties of certain complexes associated to spherical buildings. (Q1870924)

The author shows that the complex \(Opp \Delta\) of ordered pairs of opposite simplices in a spherical building \(\Delta\) is spherical, i.e., \((n-1)\) -connected, when \(n\) is the common dimension of \(\Delta\) and \(Opp \Delta\). As a corollary she obtains that \(Opp \Delta\) is even homotopy Cohen-Macaulay.