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

General

Textbook accounts:

• Stanley Kochmann, section 1.3 of Bordism, Stable Homotopy and Adams Spectral Sequences, Fields Institute Monographs 7, American Mathematical Society (1996) [ams:fim-7, cds:2264210]

• 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:PMIHES_1969__36__281_0, pdf]

• Karin Baur, Grassmannians and cluster structures, Bull. Iranian Math. Soc. 47 (2021) (Suppl 1) S5-S33 [doi:10.1007/s41980-021-00542-6]

Lecture notes:

• Michael Hopkins, Grassmannian manifolds (pdf)

See also:

• Neil Strickland, section 5 of: A Bestiary of Topological Objects [pdf, pdf]

Unstable classification of topological phases

Arguments that some effects in topological phases of matter are “unstable” or “fragile” in that the relevant deformation class of their valence bundles over the Brillouin torus is not their class in topological K-theory (as assumed by the K-theory classification of topological phases of matter) but an unstable homotopy class (what may be called a class in generalized nonabelian cohomology) such as of maps to a Grassmannian space (or more general flag variety) classifying (systems of) sub-bundles of a trivial vector bundle of fixed finite rank:

Influential precursor discussion:

• Andreas P. Schnyder, Shinsei Ryu, Akira Furusaki, Andreas W. W. Ludwig: Classification of topological insulators and superconductors in three spatial dimensions, Phys. Rev. B 78 195125 (2008) [doi:10.1103/PhysRevB.78.195125, arXiv:0803.2786]

More explicit highlighting of the role of the unstable case (and coinage of the term “fragile” for “unstable”):

• Hoi Chun Po, Haruki Watanabe, Ashvin Vishwanath: Fragile Topology and Wannier Obstructions, Phys. Rev. Lett. 121 (2018) 126402 [doi:10.1103/PhysRevLett.121.126402]

• Adrien Bouhon, Tomáš Bzdušek, Robert-Jan Slager: Geometric approach to fragile topology beyond symmetry indicators, Phys. Rev. B 102 (2020) 115135 [doi:10.1103/PhysRevB.102.115135, arXiv:2005.02044]

Applications:

• Adrien Bouhon, Robert-Jan Slager, around equation (3) in: Multi-gap topological conversion of Euler class via band-node braiding: minimal models, PT-linked nodal rings, and chiral heirs [arXiv:2203.16741]

• Zory Davoyan, Wojciech J. Jankowski, Adrien Bouhon, Robert-Jan Slager, section II.A in: Three-dimensional $\mathcal{P}\mathcal{T}$-symmetric topological phases with Pontryagin index [doi:10.1103/PhysRevB.109.165125 arXiv:2308.15555]

Expositions with an eye towards non-abelian braiding of band nodes in momentum space:

• Gunnar F. Lange: Multi-gap topology & non-abelian braiding in $\mathbf{k}$-space, talk at University of Oslo (Feb 2023) $[$pdf]

• Adrien Bouhon: Non-Abelian and Euler multi-gap topologies in crystalline materials, talk at: Quantum Information and Quantum Matter, CQTS @ NYU Abu Dhabi (May 2023) $[$pdf]

category: geometry, algebra