The Euler characteristic of a nonpositively curved, piecewise Euclidean manifold

Ruth Charney and Michael Walter Davis

Vol. 171 (1995), No. 1, 117–137

DOI: 10.2140/pjm.1995.171.117

Abstract

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 (−1)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(0) 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.

Mathematical Subject Classification 2000

Primary: 53C23

Secondary: 57M50, 57Q05

Milestones

Received: 10 January 1993

Revised: 10 August 1994

Published: 1 November 1995