On the $S^{1}$-fibred nilBott tower

Abstract

We shall introduce a notion of $S^{1}$-fibred nilBott tower. It is an iterated $S^{1}$-bundle whose top space is called an $S^{1}$-fibred nilBott manifold and the $S^{1}$-bundle of each stage realizes a Seifert construction. The $S^{1}$-fibred nilBott tower is a generalization of real Bott tower from the viewpoint of fibration. In this note we shall prove that any $S^{1}$-fibred nilBott manifold is diffeomorphic to an infranilmanifold. According to the group extension of each stage, there are two classes of $S^{1}$-fibred nilBott manifolds which is defined as finite type or infinite type. We discuss their properties.