nLab Stiefel manifold

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Contents

Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Bundles

bundles

Contents

Definition

For more context on the following cf. at classifying space.

For nn \in \mathbb{N}, write O(n)O(n) for the orthogonal group regarded as a topological group and acting canonically on n\mathbb{R}^n.

Definition

For kNk \leq N \in \mathbb{N}, the kkth real Stiefel manifold of N\mathbb{R}^N is the coset topological space.

(1)V k( N)O(N)/O(Nk), V_k\big(\mathbb{R}^N\big) \;\coloneqq\; O(N)/O(N-k) \,,

where the action of O(Nk)O(N-k) is via the standard subgroup inclusion O(Nk)O(N)O(N-k)\hookrightarrow O(N) as an upper right block.

Remark

The canonical group action O(N) NO(N) \curvearrowright \mathbb{R}^N induces a transitive action on the set of kk-dimensional linear subspaces k N\mathbb{R}^k \subset \mathbb{R}^N equipped with an orthonormal basis, and given any such subspace WW, then its stabilizer subgroup in O(N)O(N) is isomorphic to O(Nk)O(N-k), the symmetries of the orthogonal subspace W W^\perp. In this way the underlying set of V k( N)V_k(\mathbb{R}^N) is in natural bijection to the set of kk-dimensional linear subspaces in N\mathbb{R}^N equipped with orthonormal bases. The realization of this set as a coset (1) serves to equip it naturally with the structure of a topological space.

More generally, given any finite-dimensional inner product space (V,,)(V,\langle\ ,\ \rangle), there is a corresponding orthogonal group O(V)O(V), and one can repeat the above definition more or less verbatim.

Definition

By def. there are canonical inclusions V k( N)V k( N+1)V_k(\mathbb{R}^N) \hookrightarrow V_k(\mathbb{R}^{N+1}) that are compatible with the actions of the respective orthogonal groups. The colimit (in Top, see there) over these inclusions is denoted

EO(k)lim NV k( N). E O(k) \;\coloneqq\; \underset{\longrightarrow}{\lim}_N V_k(\mathbb{R}^N) \,.

This is a model for the total space of the O(k)O(k)-universal principal bundle.

Properties

Homotopy groups

Proposition

The Stiefel manifold V k( N)V_k(\mathbb{R}^N) (Def. ) is ( N k 1 ) (N-k-1) -connected.

Proof

Observe that

  1. the coset coprojection O(N)O(N)/O(Nk)O(N) \to O(N)/O(N-k) is a Serre fibration

    (by this prop. and this corollary)

  2. its fiber (hence: homotopy fiber) is O(Nk)O(N-k)

so that we have a homotopy fiber sequence of this form:

O(Nk)O(N)O(N)/O(Nk)V k( N) O(N-k) \longrightarrow O(N) \longrightarrow O(N)/O(N-k) \;\equiv\; V_k(\mathbb{R}^N)

The corresponding long exact sequence of homotopy groups has, has the following structure in degrees strictly bounded above by NkN-k, by this prop (taking n=Nkn=N-k in the notation there):

π <NkO(Nk)epiπ <NkO(N)0π <NkV k(N)0π 1<Nk1O(Nk)π 1<Nk1O(N). \cdots \to \pi_{\bullet \lt N-k} O(N-k) \overset{epi}{\longrightarrow} \pi_{\bullet \lt N-k} O(N) \overset{0}{\longrightarrow} \pi_{\bullet \lt N-k} V_k(N) \overset{0}{\longrightarrow} \pi_{\bullet-1 \lt N-k-1} O(N-k) \overset{\sim}{\longrightarrow} \pi_{\bullet-1 \lt N-k-1} O(N) \to \cdots \,.

This implies the claim. (Exactness of the sequence says that every element in π <NkV k( N)\pi_{\bullet \lt N-k} V_k(\mathbb{R}^N) is in the kernel of zero, hence in the image of 0, hence is 0 itself.)

Corollary

The colimiting space EO(k)=lim NV k( N)E O(k) = \underset{\longleftarrow}{\lim}_N V_k(\mathbb{R}^N) from Def. is weakly contractible.

Proof

This follows from Prop. , together with the fact that the sequence of maps V k( N)V k( N+1)V_k(\mathbb{R}^N) \to V_k(\mathbb{R}^{N+1}) are closed T 1T_1-inclusions. See this Prop. together with the accompanying commentary there.

CW-complex structure

Proposition

The Stiefel manifold V k( N)V_k(\mathbb{R}^N) admits the structure] of a CW-complex.

(e.g. James 1959 p. 3, James 1976 p. 5 with p. 21, Hatcher 2002 p. 302, Blaszczyk 2007)

And it should be true that with that cell structure the inclusions V k( N)V k( N+1)V_k(\mathbb{R}^N) \hookrightarrow V_k(\mathbb{R}^{N+1}) are subcomplex inclusions:

According to Yokota 1956, the inclusions SU(k)SU(N)SU(k)\hookrightarrow SU(N) are cellular such that this is compatible with the group action (reviewed here in 3.3 and 3.3.1). This implies that also the projection SU(N)SU(N)/SU(Nk)SU(N) \to SU(N)/SU(N-k) is cellular (e.g. Hatcher 2002 p. 302).

Relation to Grassmannians and universal bundles

Similarly, the Grassmannian manifold is the coset

Gr k( N)O(N)/(O(k)×O(Nk)). Gr_k(\mathbb{R}^N) \;\coloneqq\; O(N)/(O(k)\times O(N-k)) \,.

The quotient coprojection

V k( N)Gr k( N) V_k(\mathbb{R}^N) \longrightarrow Gr_k(\mathbb{R}^N)

is an O(k)O(k)-principal bundle, with associated bundle V k( N)× O(k) kV_k(\mathbb{R}^N) \times_{O(k)} \mathbb{R}^k a vector bundle of rank kk. In the limit (colimit) that NN \to \infty this gives a presentation of the O(k)O(k)-universal principal bundle and of the universal vector bundle of rank kk, respectively. The base space Gr k() wheBO(k)Gr_k(\infty)\simeq_{whe} B O(k) is the classifying space for O(k)O(k)-principal bundles and for rank kk vector bundles.

References

Named after:

  • Eduard Stiefel: Richtungsfelder und Fernparallelismus in nn-dimensionalen Mannigfaltigkeiten, Comment. Math. Helv. 8 (1935/6) 3-51

Further discussion:

See also:

Last revised on July 8, 2025 at 02:34:30. See the history of this page for a list of all contributions to it.