The topology of finite and infinite-dimensional Stiefel manifolds and the relation to Grassmannians

Abstract

Stiefel manifolds arise naturally as spaces of injective operators and as total spaces of principal bundles over Grassmannians. While their finite-dimensional topology is governed by Bott periodicity, the infinite-dimensional theory exhibits a striking collapse phenomenon stemming from Kuiper's contractibility theorem. In this expository article, we present a unified treatment of finite and infinite-dimensional Stiefel manifolds over real and complex Hilbert spaces, emphasizing three structural principles: (i) the interpretation of Stiefel manifolds as spaces of injective operators, (ii) the polar decomposition as a canonical factorization yielding a homotopy decomposition, (iii) the role or function of Grassmannians as classifying spaces for $\operatorname{GL}_n(\mathbb{F})$ in the stable limit. In particular, we show that the polar decomposition provides a global homeomorphism \[ \operatorname{St}(n,H) \cong \operatorname{St}_{\operatorname{orth}}(n,H) \times \operatorname{P}_n(\mathbb{F}) \] valid in arbitrary Hilbert dimension, and that this factorization isolates the entire difference between finite and infinite-dimensional topology. Then, we discuss the implications for the homotopy type of Stiefel manifolds and the relation of Stiefel manifolds to the theory of classifying spaces and characteristic classes.