Manchester Geometry Seminar 2003/2004

Bott towers are families of manifolds that were first invented by Bott and Samelson in the 1950s during their study of loops on Lie groups. They are complex algebraic varieties, which may be represented as iterated projective bundles, and nowadays are famous for being toric manifolds over an appropriate dimensional cube. We shall explain how their integral cohomology and complex K theory flow from this fact, and outline ways in which their real K theory may be attacked, incorporating recent work of Bahri and Benderskey on the action of the mod 2 Steenrod algebra. Applications include the enumeration of stably complex structures on Bott towers, some of which have unfamiliar properties from the algebraic geometrical viewpoint.