Homotopy Decompositions and K–theory of Bott Towers

We describe Bott towers as sequences of toric manifolds M, and identify the omniorientations which correspond to their original construction as toric varieties. We show that the suspension of M 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 [2]. By way of application we investigate stably complex structures on M, identifying those which arise from omniorientations and those which are almost complex. We conclude with observations on the rôle of Bott towers in complex cobordism theory.