The Euler characteristic of manifolds with axiom of hyperplanes

Axiom of $l$-hyperplanes is the generalization of the Cartan axiom of planes and class of Riemannian manifolds satisfying one is the prolongation of space form. The problem on defining whether the Euler characteristic is of fixed sign (positive or negative) is solved for compact manifolds with axiom of $l$-hyperplanes at enough great $l$. These Euler classes are computed explicity. Under assumption general situation for a structure curvature then great Stiefel–Whitney's class are zero. If a sign of curvature manifold $M^{2m}$ m with axiom of $(2m-2)$ hyperplanes is not definite, then $M$ is locally isometric to a direct product of a straight line and a nonflat space form.