Prove that the helicoid is homeomorphic to the plane.

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- June 7th 2010, 09:42 PMBruno J.Helicoid
Prove that the helicoid is homeomorphic to the plane.

- June 7th 2010, 10:09 PMchiph588@
- June 7th 2010, 10:12 PMBruno J.
- June 7th 2010, 10:22 PMchiph588@
- June 7th 2010, 10:34 PMBruno J.
Good!

Here are a few abstract thoughts.

We know the universal covering space of the punctured plane is the plane , with the projection .

On the other hand, the universal cover of is also the Riemann surface of the (multi-valued) inverse function , of which the helicoid is the most obvious model. By the universal property of the universal cover, there exists a homeomorphism between the helicoid and the plane.

(Informally : the slit plane is easily seen to be homeomorphic to a strip without its boundary, (any branch of the logarithm doing the job). Now lift the cut to the helicoid, and map each slice of the helicoid to a strip, identifying the appropriate boundaries of two strips if the corresponding boundaries of the slices are identified on the helicoid. In this way, the helicoid is mapped homeomorphically on the plane.)