I am interested in an example of a local homeomorphism from the open unit disc D onto itself which is not a homeomorphism. Or, could one prove that any such map is a homeomorphism?
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Use that the disc is simply connected and locally compact. If $a\ne b$ with $f(a)=f(b)$, a path from $a$ to $b$ gives a closed path from/to $f(a)$. This can be contracteded to the point $f(a)$. In the preimage this results in a path from $a$ to $b$ that is constant! – Hagen von Eitzen Mar 30 '14 at 21:44
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@HagenvonEitzen: Actually, such maps abound since they are not required to be covering maps/proper. – Moishe Kohan Mar 30 '14 at 23:30
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Still not sure...is there a local homeomorphism from D onto D/ – Dan Gallo Mar 30 '14 at 23:48
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Is there a surjective immersion from R onto R? – Dan Gallo Mar 30 '14 at 23:49
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Hagen : your proof sounds good...you are taking the preimage of the homotopy to get a constant path. It would seem this is where – Dan Gallo Mar 31 '14 at 00:09
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local compactness is needed. The proof also seems to say that – Dan Gallo Mar 31 '14 at 00:11
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any surjective immersion of R^2 to R^2 is a diffeo. – Dan Gallo Mar 31 '14 at 00:11
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1@DanGallo $f(z) = \int_0^z \exp(z^2),dz$ is a surjective immersion of $\mathbb C$ to itself. (Surjectivity follows from Picard, because the function is odd; immersion because $f'$ never vanishes). – user127096 Mar 31 '14 at 00:16
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For more details see http://mathoverflow.net/questions/147110/surjective-entire-functions-without-critical-points – Moishe Kohan Mar 31 '14 at 18:19