nLab Stiefel manifold

Definition

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

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

Definition

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

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

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

Remark

The canonical group action $O(N) \curvearrowright \mathbb{R}^N$ induces a transitive action on the set of $k$-dimensional linear subspaces $\mathbb{R}^k \subset \mathbb{R}^N$ equipped with an orthonormal basis, and given any such subspace $W$, then its stabilizer subgroup in $O(N)$ is isomorphic to $O(N-k)$, the symmetries of the orthogonal subspace $W^\perp$. In this way the underlying set of $V_k(\mathbb{R}^N)$ is in natural bijection to the set of $k$-dimensional linear subspaces in $\mathbb{R}^N$ 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 $(V,\langle\ ,\ \rangle)$, there is a corresponding orthogonal group $O(V)$, and one can repeat the above definition more or less verbatim.

Definition

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

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

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

Properties

Homotopy groups

Proposition

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

Proof

Observe that

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

   (by this prop. and this corollary)

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

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

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

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

$$\cdots \to \pi_{\bullet \lt N-k} O(N-k) \overset{epi}{\longrightarrow} \pi_{\bullet \lt N-k} O(N) \overset{0}{\longrightarrow} \pi_{\bullet \lt N-k} V_k(N) \overset{0}{\longrightarrow} \pi_{\bullet-1 \lt N-k-1} O(N-k) \overset{\sim}{\longrightarrow} \pi_{\bullet-1 \lt N-k-1} O(N) \to \cdots \,.$$

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

Corollary

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

Proof

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

CW-complex structure

Proposition

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

According to Yokota 1956, the inclusions $SU(k)\hookrightarrow SU(N)$ 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 $SU(N) \to SU(N)/SU(N-k)$ is cellular (e.g. Hatcher 2002 p. 302).

Relation to Grassmannians and universal bundles

Similarly, the Grassmannian manifold is the coset

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

The quotient coprojection

$$V_k(\mathbb{R}^N) \longrightarrow Gr_k(\mathbb{R}^N)$$

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

Related concepts

• Grassmannian manifold

• classifying space

References

Named after:

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

Further discussion:

• Yoshihiro Saito: On the homotopy groups of Stiefel manifolds, J. Inst. Polytech. Osaka City Univ. Ser. A 6 1 (1955) 39-45 [euclid:ojm/1353054734]

• 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) [doi:10.1112/plms/s3-9.1.115, pdf]

• Ioan Mackenzie James: On the homotopy type of Stiefel manifolds, Proceedings of the AMS, 29 1 (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)

• Allen Hatcher: Algebraic Topology, Cambridge University Press (2002) [ISBN:9780521795401, webpage]

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

See also:

• Wikipedia: Stiefel manifold