Topological classification of generalized Bott towers

Abstract

Abstract. If B is a toric manifold and E is a Whitney sum of complex line bundles over B, then the projectivization P (E) of E is again a toric manifold. Starting with B as a point and repeating this construction, we obtain a sequence of complex projective bundles which we call a generalized Bott tower. We prove that if the top manifold in the tower has the same cohomology ring as a product of complex projective spaces, then every fibration in the tower is trivial so that the top manifold is diffeomorphic to the product of complex projective spaces. This gives supporting evidence to what we call the cohomological rigidity problem for toric manifolds, "Are toric manifolds diffeomorphic (or homeomorphic) if their cohomology rings are isomorphic?" We provide two more results which support the cohomological rigidity problem.

Complex projective towers and their cohomological rigidity up to dimension six

“…This corollary is a generalization of the cohomological rigidity theorem for Bott manifolds up to dimension less than or equal to 6 proved in [CMS10]. Note that cohomology ring does not determine the tower structure of CP -tower.…”

“…Note that any complex vector bundles over CP 1 decomposes into a Whitney sum of line bundles. Therefore a CP -tower M ∈ M 6 2 with C 1 = CP 1 is a 2-stage generalized Bott tower, and such Bott towers are completely classified in [CMS10]. (See also [CPS].)…”

“…For example, if each complex vector bundle ξ i is a Whitney sum of complex line bundles, such CP -tower is a generalized Bott tower, introduced in [CMS10]. If each ξ i is a sum of two complex line bundles, then it is a Bott tower, introduced in [BoSa] (also see [GrKa]).…”

“…Note that if we restrict the class of 8-dimensional CP -towers to the 8-dimensional generalized Bott manifolds with height 2, then cohomological rigidity holds by [CMS10].…”

Classification problems of toric manifolds via topology

“…Simply connected real 4-dimensional closed smooth manifolds are classified up to homeomorphism by isomorphism classes of the bilinear forms defined by the intersection paring of real 2-cyclyes ( [15]), so the homeomorphism types of those manifolds are distinguished by their cohomology rings. This together with the fact that any toric manifold is smooth and simply connected ( [16 The reader can find more partial affirmative solutions to Problem 1 in [6].…”