The Euler characteristic of a nonpositively curved, piecewise Euclidean manifold.

A conjecture of H. Hopf states that if M2n is a closed, Riemannian manifold of nonpositive sectional curvature, then its Euler characteristic, χ(M2n), should satisfy (-l)nχ(M2n) > 0. In this paper, we investigate the conjecture in the context of piecewise Euclidean manifolds having "nonpositive curvature" in the sense of Gromov's CAT(O) inequality. In this context, the conjecture can be reduced to a local version which predicts the sign of a "local Euler characteristic" at each vertex. In the case of a manifold with cubical cell structure, the local version is a purely combinatorial statement which can be shown to hold under appropriate conditions. The original conjecture of Hopf, and a similar conjecture for nonnegative curvature (which we shall not be concerned with here), are true in dimensions 2 and 4, by the Chern-Gauss-Bonnet Theorem: in both cases the curvature condition forces the Gauss-Bonnet integrand to have the correct sign. This is immediate in dimension 2. Chern [Ch] gives a proof in dimension 4 and attributes the result to Milnor. A result of [Ge] shows that, in dimensions > 6, the curvature condition does not force the Gauss-Bonnet integrand to have the correct sign; hence, the same argument does not work in higher dimensions. (However, the hypothesis that the curvature operator is negative semidefinite does force the integrand to have the correct sign.)