Homotopy decompositions and \(K\)-theory of Bott towers (Q2567984)

A Bott tower of height \(n\) is a sequence of smooth orientable \(2k\)-manifolds \((M^k: k\leq n)\) determined by a list of integral \((k -1)\)-vectors. The authors begin to describe the \(k\)-stage \(M^k\) as a toric manifold and discuss the relationship with its construction as complex manifold. They 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. They also extend the results to \(KO\)-theory. As an application, they consider the enumeration of stably complex structures on \(M^k\) obtaining estimates for those which arise from omniorientations and those which are almost complex. They conclude with observations on the role of Bott towers in complex cobordism theory.