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 ${\mathrm{GL}}_n(\mathbb{F})$ 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.

14 pages