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Let $K$ be a simplicial complex whose vertices are labelled by $1,2,\cdots,k$. I want to define a variant concept of the open Stiefel manifolds $$ V_K(\mathbb{R}^n):=\{(v_1,v_2,\cdots,v_k)\in\prod_k\mathbb{R}^n\mid v_{i_1},v_{i_2},\cdots,v_{i_t}\text{ are linearly independent}\\ \text{whenever the vertices }i_1,i_2,\cdots,i_t\text{ span a simplex of } K\}. $$

Question:

(1). Has this concept been given by others before? Any name for the manifold $V_K(\mathbb{R}^n)$ and references?

(2). Are there any references for the cohomology ring $$ H^*(V_K(\mathbb{R}^n)), $$ especially with mod $p$ coefficients for $p$ prime?

(3). Could we induce any variant manifolds of Grassmannians from this $V_K(\mathbb{R}^n)$?

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