nLab
Stiefel manifold

Contents

Definition

For nn \in \mathbb{N}, write O(n)O(n) for the orthogonal group acting on n\mathbb{R}^n. For the following we regard these groups as topological groups in the canonical way.

Definition

For n,kn, k \in \mathbb{N} and nkn \leq k, then the nnth real Stiefel manifold of k\mathbb{R}^k is the coset topological space.

V n( k)O(k)/O(kn), V_n(\mathbb{R}^k) \coloneqq O(k)/O(k-n) \,,

where the action of O(kn)O(k-n) is via its canonical embedding O(kn)O(k)O(k-n)\hookrightarrow O(k).

Remark

The group O(k)O(k) acts transitively on the set of nn-dimensional linear subspaces equipped with an orthonormal basis, and given any such, then its stabilizer subgroup in O(k)O(k) is isomorphic to O(kn)O(k-n). In this way the underlying set of V n( k)V_n(\mathbb{R}^k) is in natural bijection to the set of nn-dimensional linear subspaces in k\mathbb{R}^k equipped with orthonormal basis. The realization as a coset as above serves to euqip this set naturally with a topological space.

Definition

By def. there are canonical inclusions V n( k)V n( k+1)V_n(\mathbb{R}^k) \hookrightarrow V_n(\mathbb{R}^{k+1}) that are compatible with the O(n)O(n)-action. The colimit (in Top, see there) over these inclusions is denoted

EO(n)lim kV n( k). E O(n) \coloneqq \underset{\longrightarrow}{\lim}_k V_n(\mathbb{R}^k) \,.

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

Properties

Homotopy groups

Proposition

The Stiefel manifold V n(k)V_n(k) is (n-1)-connected.

Proof

Consider the coset quotient projection

O(n)O(k)O(k)/O(n)=V n( k). O(n) \longrightarrow O(k) \longrightarrow O(k)/O(n) = V_n(\mathbb{R}^k) \,.

By this prop. and by this corollary the projection O(k)O(k)/O(n)O(k)\to O(k)/O(n) is a Serre fibration. Therefore there is the long exact sequence of homotopy groups of this fiber sequence and by this prop. it has the following structure in degrees bounded by nn:

π n1(O(k))epiπ n1(O(n))0π n1(V n(k))0π 1<n1(O(k))π 1<n1(O(n)). \cdots \to \pi_{\bullet \leq n-1}(O(k)) \overset{epi}{\longrightarrow} \pi_{\bullet \leq n-1}(O(n)) \overset{0}{\longrightarrow} \pi_{\bullet \leq n-1}(V_n(k)) \overset{0}{\longrightarrow} \pi_{\bullet-1 \lt n-1}(O(k)) \overset{\simeq}{\longrightarrow} \pi_{\bullet-1 \lt n-1}(O(n)) \to \cdots \,.

This implies the claim. (Exactness of the sequence says that every element in π n1(V n( k))\pi_{\bullet \leq n-1}(V_n(\mathbb{R}^k)) is in the kernel of zero, hence in the image of 0, hence is 0 itself.)

Corollary

The colimiting space EO(n)=lim kV n( k)E O(n) = \underset{\longleftarrow}{\lim}_k V_n(\mathbb{R}^k) from def. is weakly contractible.

CW-complex structure

Proposition

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

e.g. (James 59, p. 3, James 76, p. 5 with p. 21, Hatcher, p. 302Blaszczyk 07)

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

According to (Yokota 56) the inclusions SU(n)SU(k)SU(n)\hookrightarrow SU(k) are cellular and this is compatible with the group action (reviewed here in 3.3 and 3.3.1). This implies that also the projection SU(k)SU(k)/SU(kn)SU(k) \to SU(k)/SU(k-n) is cellular (e.g. Hatcher, p. 302).

Relation to Grassmannians and universal bundles

Similarly, the Grassmannian manifold is the coset

Gr n( k)O(k)/(O(n)×O(kn)). Gr_n(\mathbb{R}^k) \coloneqq O(k)/(O(n)\times O(k-n)) \,.

The quotient projection

V n( k)Gr n( k) V_{n}(\mathbb{R}^k)\longrightarrow Gr_n(\mathbb{R}^k)

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

References

  • Eduard Stiefel, Richtungsfelder und Fernparallelismus in

    nn-dimensionalen Mannigfaltigkeiten_, Comment. Math. Helv. , 8(1935/6), 3-51.

  • I. Yokota, On the cells of symplectic groups, Proc. Japan Acad. 32 (1956), 399-400.

  • Ioan Mackenzie James, Spaces associated with Stiefel manifolds, Proc. Lond. Math. Soc. (3) 9 (1959)

  • Ioan Mackenzie James, On the homotopy type of Stiefel manifolds, Proceedings of the AMS, vol. 29, Number 1, June 1971

  • Ioan Mackenzie James, The topology of Stiefel manifolds, Cambridge University Press, 1976

  • Stanley Kochmann, section 1.2 of Bordism, Stable Homotopy and Adams Spectral Sequences, AMS 1996

  • Zbigniew Błaszczyk, On cell decompositions of SO(n)SO(n), 2007 (pdf)

  • Hatcher, Algebraic topology

  • Wikipedia, Stiefel manifold

  • Yoshihiro Saito, On the homotopy groups of Stiefel manifolds, J. Inst. Polytech. Osaka City Univ. Ser. A Volume 6, Number 1 (1955), 39-45. Project Euclid

Last revised on May 5, 2016 at 08:26:48. See the history of this page for a list of all contributions to it.