Homotopy Decompositions and K-Theory of Bott Towers

Abstract

We describe Bott towers as sequences of toric manifolds $M^k$, and identify the omniorientations which correspond to their original construction as complex varieties. We show that the suspension of M^k is homotopy equivalent to a wedge of Thom complexes, and display its complex K-theory as an algebra over the coefficient ring. We extend the results to KO-theory for several families of examples, and compute the effects of the realification homomorphism; these calculations breathe geometric life into Bahri and Bendersky's analysis of the Adams Spectral Sequence. By way of application we consider the enumeration of stably complex structures on M^k, obtaining estimates for those which arise from omniorientations and those which are almost complex. We conclude with observations on the role of Bott towers in complex cobordism theory.