Classification of complex projective towers up to dimension 8 and cohomological rigidity

TL;DR

This paper classifies 6-dimensional complex projective towers up to diffeomorphism, establishing their cohomological rigidity, and demonstrates that this rigidity does not extend to 8-dimensional towers, providing a detailed understanding of their topological classification.

Contribution

It provides a complete classification of 6-dimensional $ ext{CP}$-towers and shows their cohomological rigidity, while also analyzing 8-dimensional cases to identify limitations.

Findings

6-dimensional $ ext{CP}$-towers are classified up to diffeomorphism

All 6-dimensional $ ext{CP}$-towers are cohomologically rigid

Cohomological rigidity fails for certain 8-dimensional $ ext{CP}$-towers

Abstract

A complex projective tower or simply a $\mathbb CP$-tower is an iterated complex projective fibrations starting from a point. In this paper we classify all 6-dimensional $\mathbb CP$-towers up to diffeomorphism, and as a consequence, we show that all such manifolds are cohomologically rigid, i.e., they are completely determined up to diffeomorphism by their cohomology rings. We also show that cohomological rigidity is not valid for 8-dimensional $\mathbb CP$-towers by classifying $\mathbb CP^1$-fibrations over $\mathbb CP^3$ up to diffeomorphism. As a corollary we show that such $\mathbb CP$-towers are diffeomorphic if they are homotopy equivalent.