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.
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.
where the action of $O(k-n)$ is via its canonical embedding $O(k-n)\hookrightarrow O(k)$.
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.
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
This is a model for the total space of the $O(n)$-universal principal bundle.
The Stiefel manifold $V_n(k)$ is (n-1)-connected.
Consider the coset quotient projection
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 $n$:
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.)
The colimiting space $E O(n) = \underset{\longleftarrow}{\lim}_k V_n(\mathbb{R}^k)$ from def. is weakly contractible.
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.
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).
Similarly, the Grassmannian manifold is the coset
The quotient projection
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.
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
Last revised on May 31, 2023 at 00:27:22. See the history of this page for a list of all contributions to it.