Moduli space of sheaves on fans

A conjecture of H. Hopf states that if x(M²n) is a closed, Riemannian manifold of nonpositive sectional curvature, then its Euler characteristic x(M²n), should satify (-1)n x(M²n)≥ 0. Ruth Charney and Michael Davis investigated the conjecture in the context of piecewise Euclidean manifolds having "nonpositive curvature" in the sense of Gromov's CAT(0) inequality. In that context the conjecture can be reduced to a local version which predicts the sign of a "local Euler characteristic" at each vertex. They stated precisely various conjectures in their paper which we are interested in one of them stated as Conjecture D (see [1]) which is equivalent to the Hopf Conjecture for piecewise Euclidean manifolds cellulated by cubes. The goal of this thesis is to study the Charney - Davis Conjecture stated as Conjecture (D) by using sheaves on fans.Science, Faculty ofMathematics, Department ofGraduat