We compute the topological simple structure set of closed manifolds that occur as total spaces of flat bundles over lens spaces S^l/(ℤ/p) with fiber Tⁿ for an odd prime p and l ≥ 3 provided that the induced ℤ/p-action on π₁(Tⁿ) = ℤⁿ is free outside the origin. To the best of our knowledge this is the first computation of the structure set of a topological manifold whose fundamental group is not obtained from torsionfree and finite groups using amalgamated and HNN-extensions. We give a collection of classical surgery invariants such as splitting obstructions and ρ-invariants that decide whether a simple homotopy equivalence from a closed topological manifold to M is homotopic to a homeomorphism. © 2020 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC

中文翻译：

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流形同伦等价于透镜空间上的某些环面束

我们计算封闭流形的拓扑简单结构集，它们作为透镜空间S ^l /(ℤ/ p )上的扁平束的总空间出现，光纤T ⁿ为奇素数p和l  ≥ 3，前提是诱导ℤ/ p -对π ₁ ( T ⁿ ) = ℤ ^{n 的作用}在原点外是免费的。据我们所知，这是第一次计算拓扑流形的结构集，其基本群不是使用合并和 HNN 扩展从无扭群和有限群中获得的。我们给出了一系列经典的手术不变量，例如分裂障碍和ρ不变量，它们决定了从封闭拓扑流形到M的简单同伦等价是否与同胚同伦。© 2020 作者。Wiley Periodicals LLC 出版的纯数学与应用数学通讯