Homotopy Decompositions and K-Theory of Bott Towers

摘要

We describe Bott towers as sequences of toric manifolds Mk, and identify the omniorientations which correspond to their original construction as complex varieties. We show that the suspension of Mk 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 Benderskys 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 Mk, obtaining estimates for those which arise from omniorientations and those which are almost complex. We conclude with observations on the r?le of Bott towers in complex cobordism theory.