Some results on the topology of real Bott towers

TL;DR

This paper investigates the topology of real Bott towers, detailing their fundamental groups, cohomology rings, and characteristic classes, and establishing conditions for properties like orientability, spin structures, and null-cobordism.

Contribution

It provides a comprehensive topological analysis of real Bott towers, including fundamental group presentations, cohomology, and combinatorial criteria for various geometric structures.

Findings

Fundamental group is abelian iff the tower is a product of circles.

Real Bott towers have solvable fundamental groups, which are nilpotent iff abelian.

The manifolds are null-cobordant with vanishing Stiefel-Whitney numbers.

Abstract

The main aim of this article is to study the topology of real Bott towers as special and interesting examples of real toric varieties. We first give a presentation of the fundamental group of a real Bott tower and show that the fundamental group is abelian if and only if the real Bott tower is a product of circles. We further prove that the fundamental group of a real Bott tower is always solvable and it is nilpotent if and only if it is abelian. We then describe the cohomology ring of a real Bott tower and also give recursive formulae for the Steifel Whitney classes. We derive combinatorial characterization for orientability of these manifolds and further give a combinatorial formula for the $(n-1)$th Steifel Whitney class. In particular, we show that if a Bott tower is orientable then the $(n-1)$th Steifel Whitney class must also vanish. Moreover, by deriving a combinatorial formula for the second Steifel-Whitney class we give a necessary and sufficient condition for the Bott tower to admit a spin structure. We finally prove the vanishing of all the Steifel-Whitney numbers and hence establish that these manifolds are null-cobordant.