Characteristic numbers, Jiang subgroup and non-positive curvature

Abstract

By refining an idea of Farrell, we present a sufficient condition in terms of the Jiang subgroup for the vanishing of signature and Hirzebruch's χy-genus on compact smooth and Kähler manifolds respectively. Along this line we show that the χy-genus of a non-positively curved compact Kähler manifold vanishes when the center of its fundamental group is non-trivial, which partially answers a question of Farrell. Moreover, in the latter case all the Chern numbers vanish whenever its complex dimension is no more than 4, which also provides some evidence to a conjecture proposed by the author and Zheng.