Topology of iterated $S^1$-bundles

TL;DR

This paper classifies the topological types of manifolds obtained as total spaces of iterated $S^1$-bundles, showing they are infra-nilmanifolds, with special focus on real Bott manifolds and their flat Riemannian structures.

Contribution

It proves that total spaces of iterated $S^1$-bundles are homeomorphic to infra-nilmanifolds and classifies their types in three dimensions, highlighting the uniqueness of real Bott manifolds.

Findings

Total spaces of iterated $S^1$-bundles are infra-nilmanifolds.

Real Bott manifolds are the only closed flat Riemannian manifolds from iterated $br{P}^1$-bundles.

Complete classification of 3-dimensional cases.

Abstract

In this paper we investigate what kind of manifolds arise as the total spaces of iterated $S^1$-bundles. A real Bott tower studied in \cite{CMO}, \cite{KM} and \cite{KN} is an example of an iterated $S^1$-bundle. We show that the total space of an iterated $S^1$-bundle is homeomorphic to an infra-nilmanifold. A real Bott manifold, which is the total space of a real Bott tower, provides an example of a closed flat Riemannian manifold. We also show that real Bott manifolds are the only closed flat Riemannian manifolds obtained from iterated $\bbr{P}^1$-bundles. Finally we classify the homeomorphism types of the total spaces of iterated $S^1$-bundles in dimension 3.