What is the topology on the Grassmannian $G_n(\mathbb{R}^{n+k})$?

Question

If the real projective space $\mathbb{P}^n=\mathbb{R}^{n+1}/(x\sim \lambda x, \lambda\in\mathbb{R})$, then how does one define a real Grassmanian $G_n(\mathbb{R}^{n+k})=\{V\subset\mathbb{R}^{n+k}\mid \dim V=n\}$ as a quotient of $\mathbb{R}^{n+k}$? I saw on Wikipedia that its topology is given by the quotient topology, without specifically referring to what quotient.

I am asking because in the definition of a tautological bundle over a Grassmannian, Wikipedia says that $G_n(\mathbb{R}^{n+k})$ is given a topology such that the map $G_n(\mathbb{R}^{n+k})\to \mathrm{End}(\mathbb{R}^{n+k})$, sending an $n$-plane $V$ to the orthogonal projection map $P_V$ onto that plane, is a homeomorphism onto its image.

https://en.wikipedia.org/wiki/Tautological_bundle#cite_note-3

As a consequence, then the set $U_V$ containing all $n$-planes $X$ such that $P_V(X)\cong X$ by a linear isomorphism is open. Is this because it is a map onto the set of automorphisms which is open from the determinant function?

Answer 1

Let $Y\subset\mathbb{R}^{n(n+k)}$ be the set of linearly independent $n$-tuples of elements of $\mathbb{R}^{n+k}$. Then there is a surjection $p:Y\to G_n(\mathbb{R}^{n+k})$ sending an element of $Y$ to its span. The topology on $G_n(\mathbb{R}^{n+k})$ is the quotient topology for this map $p$ (considering $Y$ as a subspace of $\mathbb{R}^{n(n+k)}$).

In particular, $p^{-1}(U_V)$ is the set of $n$-tuples which remain linearly independent after applying $P_V$ to each of them. As you say, this is just the set on which a certain determinant does not vanish, and so is open in $Y$.

Answer 2

Consider the collection $M$ of matrices of $n\times (n+k)$ of rank $n$ inside matrices of this size. There is a map $M \to G(n,n+k)$ that sends a matrix $x$ to its column span. This onto and induces a topology on $G(n,n+k)$. The reduced row echelon form gives you canonical representatives for the class of a matrix.

Answer 3

Taking a basis of row vectors, each element of the Grassmannian, that is a subspace of $\Bbb R^{n+k}$, can be represented as an $n\times(n+k)$ matrix of rank $n$. Of course many different matrices represent the same plane, but if $A$ is the space of full-rank $n\times(n+k)$ real matrices, it has a topology as a subspace of a vector space, and the map to the Grassmannian is a quotient map as a map of topological spaces.

Answer 4

(Just realized that the original question was asking specifically for a construction as a quotient topology. I'll still post this answer as another way to construct a topology in case this view would be useful.)

There is a canonical way to cover the Grassmannian $G_n(\mathbb{R}^{n+k})$ by $\binom{n+k}{n}$ copies of $\mathbb{R}^{kn}$, as follows: for each subset $S$ of $n$ elements of $\{ 1, \ldots, n+k \}$, we have the subspaces given by the row spaces of $n \times (n+k)$ matrices where the columns corresponding to $S$ are $e_1, \ldots, e_n$ in order. So for example, for $k=2$ and $n=3$, one possible form of matrices would be: $$ \begin{bmatrix} * & 1 & 0 & * & 0 \\ * & 0 & 1 & * & 0 \\ * & 0 & 0 & * & 1 \end{bmatrix}.$$ Each such map is injective. To see the images cover the Grassmannian, take some basis of the subspace, and then the reduced row echelon form of the matrix with these vectors as rows has the required form, and has row space equal to the original subspace.

Now, what remains is to show that on overlaps between these copies, the transition maps $(U \subseteq \mathbb{R}^{kn}) \to (V \subseteq \mathbb{R}^{kn})$ are continuous. In fact, it shouldn't be too hard to see that these transition maps are defined by rational functions.

Therefore, we can define a topology on the Grassmannian as the strong topology generated by these maps $\mathbb{R}^{kn} \to G_n(\mathbb{R}^{n+k})$, i.e. $U \subseteq G_n(\mathbb{R}^{n+k})$ is open if and only if the corresponding subsets of $\mathbb{R}^{kn}$ are each open. The condition on transition maps then implies that each copy of $\mathbb{R}^{kn}$ is homeomorphic to $\mathbb{R}^{kn}$ with its subspace topology.

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In fact, since the transition maps are also $C^\infty$, this also gives a canonical way to give the Grassmannian $G_n(\mathbb{R}^{n+k})$ a structure of $kn$-dimensional $C^\infty$ manifold.