Stiefel spaces, Grassmannians, and tautological bundles

Let $\mathbb{F}=\mathbb{R}$ or $\mathbb{C}$, and let $0\le n\le N$. The Stiefel space

$V_n({\mathbb{F}}^N)=\{(v_1,\dots ,v_n):⟨v_i,v_j⟩={\delta }_{ij}\}$

has the subspace topology from $({\mathbb{F}}^N{)}^n$. The group $K_n=\mathrm{O}(n)$ in the real case and $K_n=\mathrm{U}(n)$ in the complex case acts freely on the right by change of orthonormal frame. The Grassmannian ${\mathrm{Gr}}_n({\mathbb{F}}^N)=V_n({\mathbb{F}}^N)\mathrm{/}{K}_n$ is the space of $n$-planes with this quotient topology.

Graph charts about a plane identify nearby planes with linear maps to its orthogonal complement. They locally trivialize $V_n({\mathbb{F}}^N)\to {\mathrm{Gr}}_n({\mathbb{F}}^N)$ as a principal $K_n$-bundle. Its associated standard vector bundle, in the convention of Frame bundles and associated vector bundles, is the tautological bundle

${\gamma }_n^N=\{(W,v)\in {\mathrm{Gr}}_n({\mathbb{F}}^N)\times {\mathbb{F}}^N:v\in W\}\mathrm{.}$

Equivalently, its graph-chart transition matrices glue it by Vector bundles are glued from transition cocycles.

The coordinate inclusions ${\mathbb{F}}^N\subseteq {\mathbb{F}}^{N+1}$ define compatible inclusions of Stiefel spaces, Grassmannians, and tautological bundles. Write

$V_n({\mathbb{F}}^{\mathrm{\infty }})={\bigcup }_{N\ge n}{V}_n({\mathbb{F}}^N),{\mathrm{Gr}}_n({\mathbb{F}}^{\mathrm{\infty }})={\bigcup }_{N\ge n}{\mathrm{Gr}}_n({\mathbb{F}}^N),$

with the weak direct-limit topology: a set is closed exactly when its intersection with every finite stage is closed. The later Schubert theorem identifies this with the weak topology in CW complex with closure finiteness and weak topology. For $n=0$, the Stiefel spaces, Grassmannians, and their stable colimits are points, and ${\gamma }_0$ is the zero bundle.