nLab Stiefel manifold

Definition

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

Definition

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

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

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

Remark

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

Definition

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

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

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

Properties

Homotopy groups

Proposition

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

Proof

Consider the coset quotient projection

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

By this prop. and by this corollary the projection $O(k)\to O(k)/O(n)$ 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 $n$:

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

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

Corollary

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

Proof

This follows from the proposition, together with the fact that the sequence of maps $V_n(\mathbb{R}^k) \to V_n(\mathbb{R}^{k+1})$ are closed $T_1$-inclusions. See the Proposition here together with the accompanying commentary.

CW-complex structure

Proposition

The Stiefel manifold $V_n(\mathbb{R}^k)$ 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 $V_n(\mathbb{R}^k) \hookrightarrow V_n(\mathbb{R}^{k+1})$ are subcomplex inclusions:

According to (Yokota 56) the inclusions $SU(n)\hookrightarrow SU(k)$ 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 $SU(k) \to SU(k)/SU(k-n)$ is cellular (e.g. Hatcher, p. 302).

Relation to Grassmannians and universal bundles

Similarly, the Grassmannian manifold is the coset

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

The quotient projection

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

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

Related concepts

• Grassmannian manifold

• classifying space

References

• Eduard Stiefel, Richtungsfelder und Fernparallelismus in

  $n$-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 $SO(n)$, 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