Let $\pi$ be a group satisfying the Farrell–Jones conjecture and assume that $B\pi$ is a 4-dimensional Poincaré duality space. We consider topological, closed, connected manifolds with fundamental group $\pi$ whose canonical map to $B\pi$ has degree 1, and show that two such manifolds are s-cobordant if and only if their equivariant intersection forms are isometric and they have the same Kirby–Siebenmann invariant. If $\pi$ is good in the sense of Freedman, it follows that two such manifolds are homeomorphic if and only if they are homotopy equivalent and have the same Kirby–Siebenmann invariant. This shows rigidity in many cases that lie between aspherical 4-manifolds, where rigidity is expected by Borel’s conjecture, and simply connected manifolds where rigidity is a consequence of Freedman’s classification results.

中文翻译：

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具有 4 维基本组的拓扑 4 流形

让 $\pi$ 是一个满足 Farrell-Jones 猜想的群并假设 $B\pi$ 是一个 4 维庞加莱对偶空间。我们考虑具有基本群的拓扑、闭合、连通流形 $\pi$ 其规范映射到 $B\pi$ 度数为 1，并证明两个这样的流形是s-cobordant 当且仅当它们的等变交集形式是等距的并且它们具有相同的 Kirby-Siebenmann 不变量。如果 $\pi$ 在 Freedman 的意义上是好的，因此两个这样的流形是同胚的当且仅当它们是同伦等价的并且具有相同的 Kirby-Siebenmann 不变量。这表明在许多情况下，刚性介于非球面 4 流形之间，其中刚性是由 Borel 猜想所预期的，而简单连接流形则刚性是 Freedman 分类结果的结果。