On the \(S^1\)-fibred nilBott tower (Q2446409)

As a generalization of Bott towers, the author introduces \(S^1\)--fibred nilBott towers. These are iterated \(S^1\)--bundles, \[ M = M_n \to M_{n-1}\to \cdots \to M_2 \to M_1 \to \{pt\} \] and each stage realizes a Seifert construction. The top space is called \(S^1\)--fibered nilBott manifold. Set \(\pi_1(M_i) = \pi_i\). Associated with the fibrations there are group extensions \[ 1 \to \mathbb Z \to \pi_i \to \pi_{i-1} \to 1 \] The author proves that any \(S^1\)--fibered nilBott manifold is diffeomorphic to an infranilmanifold. According to the group entension of each stage there are two classes of \(S^1\)--fibered nilBott manifolds, those of finite and those of infinite type. Their properties are discussed in the paper.