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For $k<n$ put $FM_{k\times n}$ for the space of all $k\times n$ full rank matrices with real or complex entries. Note that the permutation group $S_{n}$ has an obvious action on this space (with permuting the columns).

On this space, consider the equivalence relation $A\sim B$ iff they have the same row spaces. Then the Grassmanian $G(k,n)$ is identified with the quotient $FM_{k\times n}/\sim$.

Does every homeomorphism of Grassmanian has a lift to a homeomorphism of $FM_{k\times n}$.

 

And is it true that this lifted homeomorphism is homotopic or isotopic to a homeomorphism on $FM_{k\times n}$ arising from a permutation of columns?

For $k<n$ put $FM_{k\times n}$ for the space of all $k\times n$ full rank matrices with real or complex entries. Note that the permutation group $S_{n}$ has an obvious action on this space (with permuting the columns).

On this space, consider the equivalence relation $A\sim B$ iff they have the same row spaces. Then the Grassmanian $G(k,n)$ is identified with the quotient $FM_{k\times n}/\sim$.

Does every homeomorphism of Grassmanian has a lift to a homeomorphism of $FM_{k\times n}$.

And is it true that this lifted homeomorphism is homotopic or isotopic to a homeomorphism on $FM_{k\times n}$ arising from a permutation of columns?

For $k<n$ put $FM_{k\times n}$ for the space of all $k\times n$ full rank matrices with real or complex entries. Note that the permutation group $S_{n}$ has an obvious action on this spce (with permuting the columns).

On this space, consider the equivalent relation $A\sim B$ iff they have the same $\textit{Row spaces}$. Then the Grassmanian $G(k,n)$ is identified with the quotion $FM_{k\times n}/\sim$.

Does every homeomorphism of Grassmanian has a lift to a homeomorphism of $FM_{k\times n}$.

And is it true that this lifted homeomorphism is homotopic or isotopic to a homeomorphism on $FM_{k\times n}$ arising from a permutation of columns?

For $k<n$ put $FM_{k\times n}$ for the space of all $k\times n$ full rank matrices with real or complex entries. Note that the permutation group $S_{n}$ has an obvious action on this space (with permuting the columns).

On this space, consider the equivalence relation $A\sim B$ iff they have the same row spaces. Then the Grassmanian $G(k,n)$ is identified with the quotient $FM_{k\times n}/\sim$.

Does every homeomorphism of Grassmanian has a lift to a homeomorphism of $FM_{k\times n}$.

And is it true that this lifted homeomorphism is homotopic or isotopic to a homeomorphism on $FM_{k\times n}$ arising from a permutation of columns?

For $k<n$ put $FM_{k\times n}$ for the space of all $k\times n$ full rank matrices? Note that the permutation group $S_{n}$ has an obvious action on this spce (with permuting the columns).

On this space, consider the equivalent relation $A\sim B$ iff they have the same $\textit{Row spaces}$. Then the Grassmanian $G(k,n)$ is identified with the quotion $FM_{k\times n}/\sim$.

Does every homeomorphism of Grassmanian has a lift to a homeomorphism of $FM_{k\times n}$.

And is it true that this lifted homeomorphism is homotopic or isotopic to a homeomorphism arising from a permutation?

For $k<n$ put $FM_{k\times n}$ for the space of all $k\times n$ full rank matrices with real or complex entries. Note that the permutation group $S_{n}$ has an obvious action on this spce (with permuting the columns).

On this space, consider the equivalent relation $A\sim B$ iff they have the same $\textit{Row spaces}$. Then the Grassmanian $G(k,n)$ is identified with the quotion $FM_{k\times n}/\sim$.

Does every homeomorphism of Grassmanian has a lift to a homeomorphism of $FM_{k\times n}$.

And is it true that this lifted homeomorphism is homotopic or isotopic to a homeomorphism on $FM_{k\times n}$ arising from a permutation of columns?