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 $V$.
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.
For $n, k \in \mathbb{N}$ and $n \leq k$, then the $n$th Grassmannian of $\mathbb{R}^k$ is the coset topological space.
where the action of the product group is via its canonical embedding $O(n)\times O(k-n) \hookrightarrow O(n)$.
Generally, for $W \subset V$ an inclusion of Euclidean vector spaces, then
where $V-W$ denotes the orthogonal complement of $W$ in $V$.
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 euqip this set naturally with a topological space.
Similarly, the real Stiefel manifold is the coset
The quotient projection
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.
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
are subcomplex incusion (hence relative cell complex inclusions).
Accordingly there is an induced CW-complex structure on the classifying space
A proof is spelled out in (Hatcher, section 1.2 (pages 31-34)).
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 $Gr(\dim V - r,V)$.
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
Lecture notes include
Last revised on January 28, 2020 at 09:39:10. See the history of this page for a list of all contributions to it.