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 $\chi_y$-genus on compact smooth and K\"{a}hler manifolds respectively. Along this line we show that the $\chi_y$-genus of a non-positively curved compact K\"{a}hler 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.