We show a rigidity theorem for the Seiberg–Witten invariants mod 2 for families of spin 4-manifolds. A mechanism of this rigidity theorem also gives a family version of 10/8-type inequality. As an application, we prove the existence of non-smoothable topological families of 4-manifolds whose fiber, base space, and total space are smoothable as manifolds. These non-smoothable topological families provide new examples of $4$-manifolds $M$ for which the inclusion maps $\operatorname {Diff}(M) \hookrightarrow \operatorname {Homeo}(M)$ are not weak homotopy equivalences. We shall also give a new series of non-smoothable topological actions on some spin $4$-manifolds.

我们显示了自旋4流形族的Seiberg-Witten不变量mod 2的刚性定理。该刚性定理的机制还给出了10/8型不等式的族形式。作为一个应用，我们证明了4流形的非光滑拓扑族的存在，这些流形的纤维，基础空间和总空间可以作为流形平滑。这些不可平滑的拓扑族提供了$ 4 $-流形$ M $的新示例，对于这些示例，包含映射$ \ operatorname {Diff}（M）\ hookrightarrow \ operatorname {Homeo}（M）$并不是弱的同伦等价物。我们还将对某些旋转的$ 4 $流形给出一系列新的不平滑拓扑动作。