Infinite Grassmannians

Definition 9.2.1 (Infinite grassmannian). Let \(\C ^{\infty }\) denote the colimit in \(\Top \) of the complex vector spaces \(\C ^n\) along the standard inclusions \(\C ^n\hookrightarrow \C ^{n+1}\). For \(n\geq k\), let \(\Gr _k(\C ^n)\) be the Grassmannian of \(k\)-dimensional linear subspaces of \(\C ^n\), with its usual topology. We define the infinite Grassmannian of \(k\)-planes as \[ \Gr _k(\C ^{\infty }) := \bigcup _{n\geq k}\Gr _k(\C ^n), \] equipped with the colimit topology: a subset \(U\subseteq \Gr _k(\C ^\infty )\) is open if and only if \(U\cap \Gr _k(\C ^n)\) is open for every \(n\).

Similarly, let \(V_k(\C ^n)\) be the Stiefel manifold of orthonormal \(k\)-frames in \(\C ^n\). The infinite Stiefel manifold \[ V_k(\C ^{\infty }) := \bigcup _{n\geq k}V_k(\C ^n) \] is likewise given the colimit topology. Its points may equivalently be regarded as linear isometric embeddings \(\C ^k\hookrightarrow \C ^\infty \). Sending a frame to the subspace it spans defines a surjective map \[ \pi \colon V_k(\C ^{\infty }) \to \Gr _k(\C ^{\infty }), \qquad (\phi \colon \C ^k \hookrightarrow \C ^{\infty }) \mapsto \phi (\C ^k), \] whose restriction at every finite stage is the usual quotient map \(V_k(\C ^n)\to \Gr _k(\C ^n)\).