In this paper we study the mod 2 cohomology ring of the Grasmannian \(\widetilde{G}_{n,3}\) of oriented 3-planes in \({\mathbb {R}}^n\). We determine the degrees of the indecomposable elements in the cohomology ring. We also obtain an almost complete description of the cohomology ring. This description allows us to provide lower and upper bounds on the cup length of \(\widetilde{G}_{n,3}\). As another application, we show that the upper characteristic rank of \(\widetilde{G}_{n,3}\) equals the characteristic rank of \(\widetilde{\gamma }_{n,3}\), the oriented tautological bundle over \(\widetilde{G}_{n,3}\) if n is at least 8.

中文翻译：

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定向三平面格拉斯曼方程的同调环和上特征秩

在本文中，我们研究了\（{\ mathbb {R}} ^ n \）中定向3平面的Grasmannian \（\ widetilde {G} _ {n，3} \）的mod 2同调环。我们确定了同调环中不可分解元素的程度。我们还获得了对同调环的几乎完整的描述。此描述使我们能够提供杯长\（\ widetilde {G} _ {n，3} \）的上限和下限。作为另一个应用程序，我们证明\（\ widetilde {G} _ {n，3} \）的特征等级等于\（\ widetilde {\ gamma} _ {n，3} \）的特征等级，如果n至少为8 ，则\（\ widetilde {G} _ {n，3} \）上的定向重言式束。