Genus in higher dimensions

Discussion of shapes with curves and holes in various dimensions.

Genus in higher dimensions

Postby Plasmath » Thu May 13, 2021 6:10 pm

In three dimensions, you can classify how many holes there are in a 3D object with its genus (the number of 3D torii attached) with a single number.
I was wondering if in four dimensions, would you need four different genuses (genii?) to classify a 4D object (one for the spheritorus, one for the torisphere, one for the ditorus, and one for the tiger)?
Or is it different? Maybe you need less or more genuses? Or are they all equivalent?
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Re: Genus in higher dimensions

Postby ICN5D » Thu May 13, 2021 10:05 pm

I don't know the correct, formal answer here. But, it is my suspicion that genus is extendable to any number of dimensions. By taking a torus of genus-1 and revolving it into 4D to generate a new ring-like object with an additional hole, seems like it would add up to a genus-2 surface in 4D (be it the tiger or ditorus). Then, you can take that genus-2 object and revolve yet again, adding another hole, forming a genus-3 surface in 5D (be it the ((((II)I)I)I) , (((II)(II))I) , or (((II)I)(II)) ). That's what I'm seeing in my head, at least. Could be wrong, though! Great question for mathematics stack exchange.
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Re: Genus in higher dimensions

Postby Marek14 » Thu Jul 15, 2021 7:11 am

I'd say that if you divide the surface into cells, genus will be the term that expresses the discrepancy from Euler characteristic.
In 4D, 3-sphere would have characteristic 0. Let's take a pentachoron as an example:

5 cells
10 faces
10 edges
5 vertices

5 - 10 + 10 - 5 = 0
C - F + E - V = 0

This holds for every convex polytope.

If you make a polytope on a surface of a toratope instead of a sphere, its Euler characteristic will be different. I think it would always be an even number and genus will be half of that number?
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