In this paper, we prove various results on the topology of the Grassmannian of oriented 3-planes in Euclidean 6-space and compute its cohomology ring. We give self-contained proofs. These spaces come up when studying submanifolds of manifolds with calibrated geometries. We collect these results here for the sake of completeness. As applications of our algebraic topological study we present some results on special Lagrangian-free embeddings of surfaces and 3-manifolds into the Euclidean 4 and 6-space.

中文翻译：

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特殊拉格朗日流形的代数拓扑

在本文中，我们证明了欧几里得 6 空间中定向 3 平面的格拉斯曼拓扑的各种结果并计算其上同调环。我们提供独立的证明。这些空间是在研究具有校准几何形状的流形的子流形时出现的。为了完整起见，我们在此收集这些结果。作为我们的代数拓扑研究的应用，我们展示了一些关于表面和 3 流形到欧几里得 4 和 6 空间的特殊无拉格朗日嵌入的结果。