As of today (31st of October, 2024), the Wikipedia article on the Classifying space for $SU(n)$ contains the term
complex oriented Grassmannians.
I was wondering what that is. Comparing with the real case, and with the application of universal bundles in mind, I would expect it to be an $S^1$-bundle $$\tilde G(k, \mathbb C^n) \to G(k, \mathbb C^n).$$ I guess I could just define it as $$\tilde G(k, \mathbb C^n) = V_n^{n+k} / SU(n),$$ where $$V_n^{n+k} = \{(v_1, \dotsc, v_n) \text{ tuple of } n \text{ orthonormal vectors in } \mathbb C^{n+k}\}$$ is the complex Stiefel manifold. But is there a somewhat more conceptional description?
Searching the web did not turn up anything. Comparing the aforementioned article with the one on Classifying space for $SO(n)$ made me wonder if someone just copied the article for $SO(n)$ and replaced every $SO(n)$ by $SU(n)$ and any mention of $\mathbb R$ by $\mathbb C$. They didn't even make a good job, because both articles contain the phrase
Since real oriented Grassmannians can be expressed as a homogeneous space by [...]
(Emphasis by me).
Any reference would be welcome.
