Title: Bordism of projective space bundles and geometric applications

Abstract: We construct special basis sequences for both the oriented 
and the complex rational cobordism rings arising from the 
consideration of complex projective space bundles over four-manifolds. 
The existence of these special basis sequences has several geometric 
applications. In the case of complex cobordism it leads to a complete 
solution of a long-standing problem posed by Hirzebruch, who asked 
which linear combinations of Chern numbers are topological invariants 
of smooth complex projective varieties. In the oriented case we find 
new characterizations of the signature, some topological and some 
differential-geometric. The latter are related to the geometry of 
nonnegatively curved manifolds.