nLab Grassmannian




topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory


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


For nn \in \mathbb{N}, write O(n)O(n) for the orthogonal group acting on n\mathbb{R}^n. More generally, say that a Euclidean vector space VV is an inner product space such that there exists a linear isometry V nV \overset{\simeq}{\to} \mathbb{R}^n to n\mathbb{R}^n equipped with its canonical inner product. Then write O(V)O(V) for the group of linear isometric automorphisms of VV. For the following we regard these groups as topological groups in the canonical way.


For n,kn, k \in \mathbb{N} and nkn \leq k, then the nnth Grassmannian of k\mathbb{R}^k is the coset topological space.

Gr n( k)O(k)/(O(n)×O(kn)), 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)×O(kn)O(k)O(n)\times O(k-n) \hookrightarrow O(k).

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

Gr W(V)O(V)/(O(W)×O(VW)), Gr_W(V) \coloneqq O(V)/(O(W)\times O(V-W)) \,,

where VWV-W denotes the orthogonal complement of WW in VV.


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


Relation to Stiefel manifolds and universal bundles

Similarly, the real Stiefel manifold is the coset

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

The quotient projection

V n(k)Gr n(k) V_{n}(k)\longrightarrow Gr_n(k)

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

CW-complex structure


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

Gr n( k)Gr n( k+1) 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

BO(n)lim kGr n( k). 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)).


(complex projective space is Oka manifold)
Every complex projective space P n\mathbb{C}P^n, nn \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)



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

  • for r=0r = 0: the point;

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

  • for r=n+1r = n+1: the point;

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

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


Textbook accounts include

Lecture notes include

category: geometry, algebra

Last revised on December 12, 2022 at 19:51:32. See the history of this page for a list of all contributions to it.