nLab Grassmannian

Idea

For $V$ a vector space and $r$ a cardinal number (generally taken to be a natural number), the Grassmannian $Gr(r,V)$ is the space of all $r$-dimensional linear subspaces of $V$.

Definition

For $n \in \mathbb{N}$, write $O(n)$ for the orthogonal group acting on $\mathbb{R}^n$. More generally, say that a Euclidean vector space $V$ is an inner product space such that there exists a linear isometry $V \overset{\simeq}{\to} \mathbb{R}^n$ to $\mathbb{R}^n$ equipped with its canonical inner product. Then write $O(V)$ for the group of linear isometric automorphisms of $V$. 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 Grassmannian of $\mathbb{R}^k$ is the coset topological space.

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

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

Generally, for $W \subset V$ an inclusion of Euclidean vector spaces, then

$$Gr_W(V) \coloneqq O(V)/(O(W)\times O(V-W)) \,,$$

where $V-W$ denotes the orthogonal complement of $W$ in $V$.

Remark

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

Properties

Relation to Stiefel manifolds and universal bundles

Similarly, the real Stiefel manifold is the coset

$$V_n(k) \coloneqq O(k)/O(k-n) \,.$$

The quotient projection

$$V_{n}(k)\longrightarrow Gr_n(k)$$

is an $O(n)$-principal bundle, with associated bundle $V_n(k)\times_{O(n)} \mathbb{R}^n$ 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.

CW-complex structure

Proposition

The real Grassmannians $Gr_n(\mathbb{R}^k)$ and the complex Grassmannians $Gr_n(\mathbb{C}^k)$ admit the structure of CW-complexes. Moreover the canonical inclusions

$$Gr_n(\mathbb{R}^k) \hookrightarrow Gr_n(\mathbb{R}^{k+1})$$

are subcomplex incusion (hence relative cell complex inclusions).

Accordingly there is an induced CW-complex structure on the classifying space

$$B O(n) \simeq \underset{\longrightarrow}{\lim}_k Gr_n(\mathbb{R}^k) \,.$$

A proof is spelled out in (Hatcher, section 1.2 (pages 31-34)).

Proposition

(complex projective space is Oka manifold)
Every complex projective space $\mathbb{C}P^n$, $n \in \mathbb{N}$, is an Oka manifold. More generally every Grassmannian over the complex numbers is an Oka manifold.

(review in Forstnerič & Lárusson 11, p. 9, Forstnerič 2013, Ex. 2.7)

Examples

Example

The Grassmannian $Gr_r(\mathbb{R}^{n+1})$ (def. ) is

• for $r = 0$: the point;

• for $r = 1$: the real projective space $\mathbb{R}P^n$;

• for $r = n+1$: the point;

• for $r \gt n+1$: the empty space $\emptyset$.

If $V$ is an inner product space, then the orthogonal complement defines an isomorphism between $Gr(r,V)$ and $Gr(\dim V - r,V)$.

Related concepts

• isotropic Grassmannian

• Lagrangian Grassmannian, affine Grassmannian

• flag variety, Schubert variety

• Stiefel manifold

• coset space

• projective space

• classifying space, universal bundle

References

Textbook accounts include

• Stanley Kochmann, section 1.3 of Bordism, Stable Homotopy and Adams Spectral Sequences, AMS 1996

• Allen Hatcher, section 1.2 of Vector bundles & K-theory (web)

• SL Kleiman, Geometry on Grassmannians and applications to splitting bundles and smoothing cycles, Publications Mathématiques de l’IHÉS, 1969, numdam/pdf

• Karin Baur, Grassmannians and cluster structures, Bull. Iranian Math. Soc. 47 (2021) (Suppl 1):S5–S33 doi

Lecture notes include

• Michael Hopkins, Grassmannian manifolds (pdf)

category: geometry, algebra