# Diffeomorphism: surface x4 + y6 + z2 = 1

I'm trying to figure out what the surface x4 + y6 + z2 = 1 looks like.
I want to say that it is diffeomorphic to the sphere because (x2)2 + (y3)2 + (z)2 = 1
but i can't seems to actually construct the diffeomorphism (I am having problems with the x2 being invertible).
Please let me know if i'm on the right track (if i'm even right)

I tried (x,y,z)|-->(x2,y3,z) but then only positive x are mapped to (and twice)
So then I tried (x,y,z)|-->(sgn(x)x2,y3,z) which is bijective but isn't smooth (it doesn't have a second derivative when x=0)
Any suggestions?

Hurkyl
Staff Emeritus
Gold Member

You could just graph it. What are its cross sections in planes parallel to the x-y plane?

Or, you could just use your mapping to understand the points away from the points where where x=0, and find some other means of understanding the subspace of points of small x.

What is the application? Is it really not enough to simply have a homoeomorphism?

P.S. my instinct is to map radially from the origin. Descartes rule of signs proves this is well defined for all points not lying in a coordinate plane.,,,

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I was hoping to find the euler characteristic for it, so i suppose a homeomorphism would be sufficient. In this case my second construction would be suitable correct?
I graphed the level curves and they are rounded off squares which is much like i anticipated. (also it suggests your idea of mapping out radially would probably work)

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