Grassmannian as classifying space

Example

Let $n \ge 1$. For $k \ge n$, let $G_n(\R^k)$ be the Grassmannian of $n$-dimensional linear subspaces of $\R^k$. The inclusions $\R^k \subset \R^{k+1}$ give inclusions $G_n(\R^k) \subset G_n(\R^{k+1})$, and $G_n(\R^\infty) = \bigcup_k G_n(\R^k)$ carries the weak topology of this union. The tautological bundle $$\gamma^n = \set{(V, v) \in G_n(\R^\infty) \times \R^\infty : v \in V}$$ is a real vector bundle of rank $n$ over $G_n(\R^\infty)$.

Let $X$ be a paracompact space. Then $[f] \mapsto f^* \gamma^n$ is a bijection from homotopy classes of maps $X \to G_n(\R^\infty)$ to isomorphism classes of real rank $n$ bundles over $X$. Rank $n$ real vector bundles correspond to principal $O(n)$-bundles through their bundles of orthonormal frames. So $G_n(\R^\infty)$ is a classifying space $BO(n)$ of the orthogonal group $O(n)$. The frame bundle of $\gamma^n$ is the universal bundle; its total space is the infinite Stiefel manifold $V_n(\R^\infty)$, which is contractible.