# Homomorphism between the set of n-flats in $R^m$ to some manifold

I am unaware of what the formal definition of "limit" is for a sequence of flats, but for the purpose of this question her is the definition of limit that I am using: Consider a sequence $s_1, s_2, s_3,$ etc. where $s_i$ is a flat of $R^m$ for all $i$. We will say that $lim_{n\to\infty}s_n=S$ if and only if for any point $p$ on $S$ and any positive value $\epsilon$, there exists a sufficiently large number $b$ such that if $x>= b$, the minimum distance between $p$ and $s_x$ will be no greater than $\epsilon$.

Here is my question:

1) Does there exist a homomorphism between the set of all n-flats in $R^m$ to some manifold, for all n and m, and 2) If so, to what manifold?

This question is partially motivated by a different question I asked earlier, where it was established that the set of all lines in a plane could not be continuously and bijectively represented by the set of all points in the plane, because the set of lines in a plane is homomorphic to an open mobius strip. Continuous bijective way of representing a line on a plane [answered]

Note: I know very little topology, so I would appreciate it if people provided links if they refer to even "well known" theories.

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I think you want the Grassmannian of n-planes in m-space. Wikipedia provides some background. – Hugh Thomas Jan 10 '11 at 23:11
@hugh Thomas This is close, but the grasmannian is related to linear subspaces, not the set of all flats of a certain dimension. So, any flat that doesn't pass through the origin would be excluded. – yrudoy Jan 10 '11 at 23:23
If you want affine subspaces of dimension n in R^m, you can fix a copy Z of R^m inside R^{m+1} which doesn't pass through the origin. Now \cap Z provides a map from an open subset of the Grassmannian of n+1-dimensional linear subspaces of R^{m+1}, to n-dimensional affine subspaces of Z=R^m. – Hugh Thomas Jan 11 '11 at 0:44
@hugh Thomas: I agree that you can use this approach to show that the answer to my question is "an open subset of the Grassmannian of n+1-dimensional linear subspaces of R^{m+1}", but I don't believe this uniquely identifies a manifold. – yrudoy Jan 11 '11 at 15:21
The space of n-flats in $R^m$ is sometimes called an affine Grassmannian. Wikipedia gives two different definitions for this term, one of which seems to be what you want. The affine Grassmannian in this sense contains the ordinary Grassmannian and is a vector bundle over it, so it is a manifold but it is not homeomorphic to a euclidean space since it has the same homology and homotopy groups as the ordinary Grassmannian, and many of these groups are nontrivial, starting with the fundamental group. – Allen Hatcher Jan 11 '11 at 16:37