Complex tautological bundle on a Grassmannian
= Complex tautological bundle on a Grassmannian
{title2=$\gamma_k^N=\{(W,v):v\in W\}$}
Over $\operatorname{Gr}_k(\mathbb C^N)$ this rank-k <complex vector bundle> has fibre W at the point W. In a chart of planes that are graphs of maps from a fixed k-plane, $(A,w)\mapsto(\operatorname{graph}A,w+Aw)$ gives a <local trivialization>. The direct-limit bundle over the infinite <Grassmannian> classifies rank-k bundles over compact Hausdorff spaces.