Topology of iterated $S^{1}$-bundles

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 [2], [13] and [14] 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 $\mathbb{R}P^{1}$-bundles. Finally we classify the homeomorphism types of the total spaces of iterated $S^{1}$-bundles in dimension 3.