  We shall introduce a notion of $S^1$-fibred nilBott tower. It is an iterated
$S^1$-bundles whose top space is called an $S^1$-fibred nilBott manifold and
the $S^1$-bundle of each stage realizes a Seifert construction. The 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.

Algebraic Topology