On the $S^1$-fibred nil-Bott Tower

TL;DR

This paper introduces $S^1$-fibred nilBott towers, generalizing real Bott towers, and proves that such manifolds are diffeomorphic to infranilmanifolds, exploring their properties based on group extension types.

Contribution

It defines $S^1$-fibred nilBott towers and proves their diffeomorphism to infranilmanifolds, extending the understanding of fibration structures in geometric topology.

Findings

Any $S^1$-fibred nilBott manifold is diffeomorphic to an infranilmanifold.

Classification into finite and infinite type based on group extension.

Properties depend on the group extension type.

Abstract

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.