2005.34: Homotopy Decompositions and K-theory of Bott Towers

2005.34: Yusuf Civan and Nigel Ray (2004) Homotopy Decompositions and K-theory of Bott Towers. K-theory, 34 (1). pp. 1-33. ISSN 0920-3036

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DOI: 10.1007/s10977-005-1551-x

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.

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