Results on the topology of generalized real Bott manifolds

TL;DR

This paper extends the topological analysis of real Bott manifolds to generalized real Bott manifolds, providing fundamental group presentations, homotopy group calculations, and criteria for orientability and spin structures.

Contribution

It offers a comprehensive topological characterization of generalized real Bott manifolds, including fundamental group properties and conditions for orientability and spin structures.

Findings

Fundamental group is solvable and conditions for it to be abelian are given.

Manifolds are aspherical only in the case of real Bott manifolds.

Higher homotopy groups are computed for these manifolds.

Abstract

Generalized Bott manifolds (over $\mathbb C$ and $\mathbb R$) have been defined by Choi, Masuda and Suh. In this article we extend the results of arXiv:1609.05630 on the topology of real Bott manifolds to generalized real Bott manifolds. We give a presentation of the fundamental group, prove that it is solvable and give a characterization for it to be abelian. We further prove that these manifolds are aspherical only in the case of real Bott manifolds and compute the higher homotopy groups. Furthermore, using the presentation of the cohomology ring with $\mathbb Z_2$-coefficients, we derive a combinatorial characterization for orientablity and spin structure.