Is it possible to continuously choose one-dimensional subspace in each k-dimensional subspace?

Question

Does there exist a continuous map from Grassmann manifold to projective space $Gr^n(V) \to \mathbb P(V)$, such that image of every n-dimensional subspace lies (1-dimensional subspace) in this subspace? I mean that we choise 1-dimensional subspace in each n-dimensional space continuously. Or, at least, in the simple cases $Gr^2(\mathbb R^3) \to \mathbb P(\mathbb R^3)$ and $Gr^2(\mathbb R^4) \to \mathbb P(\mathbb R^4)$?

Answer 1

The existence of one dimensional subbundle $E$ of n-dimensional tautological bundle $\tau(Gr^n(V))$ implies $\tau(Gr^n(V))=E+E^\perp$ and the top Stiefel-Whitney class $w_n(\tau(Gr^n(V)))=0$. But there is the following diagram

$$\begin{array}{ccl} \tau&\longrightarrow&\tau&=&EO(n)\\ \downarrow&&\downarrow&&\;\;\;\downarrow\\ Gr^n(V)&\stackrel{i}\longrightarrow&Gr^n(\infty)&=&BO(n)\\ \end{array}$$

inducing

$$\begin{array}{ccl} H^n(BO(n))&\stackrel\sim\longrightarrow&H^n(Gr^n(V))\\ w_n(EO(n))&\mapsto&w_n(\tau(Gr^n(V)))&=&0 \end{array}$$

so $w_n(EO(n))=0$ and all n-dimensional bundles should have the top Stiefel-Whitney class 0, contradiction.