The theory of intersection spaces assigns cell complexes to certain stratified topological pseudomanifolds depending on a perversity function in the sense of intersection homology. The main property of the intersection spaces is Poincaré duality over complementary perversities for the reduced singular (co)homology groups with rational coefficients. This (co)homology theory is not isomorphic to intersection homology, instead they are related by mirror symmetry. Using differential forms, Banagl extended the intersection space cohomology theory to 2-strata pseudomanifolds with a geometrically flat link bundle. In this paper, we use differential forms on manifolds with corners to generalize the intersection space cohomology theory to a class of 3-strata spaces with flatness assumptions for the link bundles. We prove Poincaré duality over complementary perversities for the cohomology groups. To do so, we investigate fiber bundles on manifolds with boundary. At the end, we give examples for the application of the theory.

中文翻译：

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三层赝流形的交叉空间上同调

交叉空间理论根据交叉同源意义上的反常函数将细胞复合物分配给某些分层的拓扑伪流形。交集空间的主要性质是具有有理系数的简化奇异（共）同调群的互补反常上的庞加莱对偶。这种（共）同调理论与交叉同调不是同构的，而是它们通过镜像对称相关。使用微分形式，Banagl 将交叉空间上同调理论扩展到具有几何扁平连接束的 2 层赝流形。在本文中，我们使用带角流形上的微分形式，将交集空间上同调理论推广到一类具有连接束平坦度假设的三层空间。我们证明了上同调群互补反常的庞加莱对偶。为此，我们研究了具有边界的流形上的纤维束。最后，我们举例说明了该理论的应用。