ON A GENERALIZATION OF HIRZEBRUCH'S THEOREM TO BOTT TOWERS

Abstract

The primary aim of this paper is to generalize a theorem of Hirzebruch for the complex 2-dimensional Bott manifolds, usually called Hirzebruch surfaces, to more general Bott towers of height n. To do so, we first show that all complex vector bundles of rank 2 over a Bott manifold are classified by their total Chern classes. As a consequence, in this paper we show that two Bott manifolds Bn(�1,...,�n−1,�n) and Bn(�1,...,�n−1,�') are isomorphic to each other, as Bott towers if and only if bothn � � ' mod 2 and � 2 = (� ') 2 hold in the cohomology ring of Bn−1(�1,...,�n−1) over integer coefficients. This result will complete a circle of ideas initiated in (11) by Ishida. We also give some partial affirmative remarks toward the assertion that under certain c our main result still holds to be true for two Bott manifolds just diffeomorphic, but not necessarily isomorphic, to each other.