The Chern conjecture for affinely flat manifolds using combinatorial methods (Q1397898)

A well known conjecture of Chern asserts that a compact affinely flat manifold \(M^n\) has vanishing Euler characteristic. In this paper the author proves this conjecture when \(M^n\) decomposes along codimension one submanifolds with finite holonomy into n-dimensional pieces with amenable holonomy. The conjecture had been proved by \textit{M. W. Hirsch} and \textit{W. P. Thurston} [Ann. Math. (2) 101, 369-390 (1975; Zbl 0321.57015)] when the holonomy group is build up out of amenable groups, by taking free products and finite extensions. The author uses a polyhedral Gauss Bonnet formula as in \textit{H. Kim} and \textit{H. Lee} [Topology Appl. 40, 195-201 (1991; Zbl 0727.57022)].