I'm not sure what you mean, the domain has to be compact, so I don't think a patch job is helpful, unless I'm misunderstanding you (which is entirely possible). If that was another misunderstanding, please comment to that effect, otherwise I'll try and think about what you said to figure out how to work it.
The problem with
is that it fails to inject when t>0, and it manifests in the example I gave in my previous post, namely: the map
near 0 (and infinity) is
, and so if you try to pull apart with nonzero derivative at 0 and infinity, you end up with this same problem.
Of course, you can measure your "smallness" in a different norm that accounts for that (essentially, C^1 with respect to another differential structure such that your original map is an immersion).
Right, the issue is that we were supposed to have a compact domain, and ostensibly this might not be a problem there.
I made a small error before, though, my domain and codomain should be [-1,1], and then I can break all three at once I think, since
is a smooth homeomorphism on
and for small t, I know that I can guarantee that the
still has the right codomain (the extreme values on the interior happen at
, and give outputs which are within the right range to still land in [-1,1]. Since I break 1-1 AND onto, clearly the homeomorphism property is also broken.
I like the projective space version as well, it gives me a boundary-less version, which is really nice, because I could not think of one before that.
Thanks for the help outermeasure, and for the expanded ideas/examples--they're quite understandable, and nice to have in my box of things to draw on!