On Grassmann manifold

In most references about Grassmann manifold, we usually introduce the following map: suppose that $$S$$ is a fixed subspace of codimension $$k$$, and let $$\Omega_S \subset Gr(k,\mathbb{K})$$ be all subspaces $$W$$ such that $$W \cap S = \{0\}$$. $$\Omega_S$$ is an open set.

Fix some $$U \in \Omega_S$$. One can show that $$W \in \Omega_S$$ if and only if there exists some linear map $$\beta_W \in Hom(U,S)$$ such that $$W = \{x + \beta_W x : x \in U \}$$. From there, one can easily check that there exists a bijection $$\Omega_S \simeq Hom(U,S)$$.

I'm interested in show that this bijection is actually an homeomorphism of topological spaces!

So far, I haven't been able to make a lot about this claim. What bugs me is that the topologies at stake are not necessarily intuitive to work with: $$Gr(r, \mathbb{K})$$ is endowed with the quotient topology, and $$Hom(U,S)$$ is really just an euclidean space, so might just take some random norm/distance.

Any help greatly appreciated !

EDIT : Look here Lee's page 22 for more informations.

• But Lee uses these bijections to define the topology on the Graßmannian, so you should be more explicit about which topology you choose on it. Since you said 'quotient topology' I assume you mean the construction as a homogeneous space en.m.wikipedia.org/wiki/… ? – Ben Oct 31 '18 at 9:55

This is becoming old, but anyway, I was able to find an answer myself and never posted it here, so here we go. Define $$t : \Omega_S \to Hom(U, S).$$ as above. Let $$\pi : V_0^k \to Gr(k, \mathbb{K})$$ where $$V_0^k := \{ (v_1, \dots, v_k) : v_i\text{'s linearly independent} \}$$, with $$\pi(v_1, \dots, v_k) = Vect(v_1, \dots, v_k),$$defines the quotient topology over $$Gr(k, \mathbb{K})$$, in which $$\Omega_S$$ is open. By definition, it suffices to show that $$t' : \pi^{-1}(\Omega_S) \to Hom(U,S)$$ in continuous, where $$t'(v_1, \dots, v_k) = -pr_{S/Vect(v_1, \dots, v_k)}$$ ($$pr_{A/B}$$ is the projection on $$A$$ w.r.t. to $$B$$). Let $$w_1, \dots, w_{n-k}$$ be a base for $$S$$. Define the (clearly) continuous application $$A : \pi^{-1}(\Omega_S) \ni (v_1, \dots, v_k) \mapsto [v_1, \dots, v_k, w_1, \dots, w_{n-k} ] \in \mathcal{M}_{n}(\mathbb{K})$$ (i.e. the matrix with columns $$v_i's$$ and $$w_i's$$). Then $$t'(v_1, \dots, v_k) = \left( x \mapsto \sum_{i=1}^{n-k} \left\langle x, (A(v_1, \dots, v_k)A^t(v_1, \dots, v_k))^{-1}w_i \right\rangle w_i\right)$$which is easily seen to be continuous.