Homotopy decompositions and K-theory of Bott towers

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 [Bahri, A. and Bendersky, M.: The KO-theory of toric manifolds. Trans. Am. Math. Soc. 352 (2000), 1191-1202.] 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 rôle of Bott towers in complex cobordism theory. © Springer 2005.