Title: The Euler characteristic of a nonpositively curved, piecewise Euclidean manifold
Abstract:A conjecture of H. Hopf states that if M 2n is a closed, Riemannian manifold of nonpositive sectional curvature, then its Euler characteristic, χ(M 2n ), should satisfy (-l) n χ(M 2n ) > 0. In this pa...A conjecture of H. Hopf states that if M 2n is a closed, Riemannian manifold of nonpositive sectional curvature, then its Euler characteristic, χ(M 2n ), should satisfy (-l) n χ(M 2n ) > 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.Read More