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.

  • $\begingroup$ 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/… ? $\endgroup$ – Ben Oct 31 '18 at 9:55

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