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Does anyone know a reference to the classifications of local homeomorphisms from a closed line segment into $\mathbb{R^2}$ ? I suspect it is given by the minimal number of self intersections of the image curve.

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    Classification up to a global homeomorphism of $\mathbb R^2$?2012-06-07
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    If the map is given by f:[a,b]\to R^2, I am interested in a classification up to homotopy (in R^2) with endpoints f(a) and f(b) fixed.2012-06-07
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    Then my guess is that you can homotope anything into the straight line segment, though preserving the local homeomorphism property along the way will give a bit of headache.2012-06-07
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    Thanks. You are correct. I've got to rethink my question.2012-06-07

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