A combinatorial criterion for the vanishing of Pontryagin classes (Q1313948)

For any given cellular decomposition of a manifold \(M\) the author obtains a local combinatorial criterion which implies that the Pontryagin classes \(P_1 (M), \dots, P_r (M)\) simultaneously vanish. This criterion involves hypersimplicial systems and the projectability of Gaussian configuration systems. It is not explicitly indicated how to test the criterion for a concrete example; compare a recent paper by \textit{L. Milin} [A combinatorial computation of the first Pontryagin class of the complex projective plane, Geom. Dedicata 49, No. 3, 253-291 (1994)].