Some results on the topology of real Bott towers (1609.05630v1)

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.