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July 5th, 2019, 12:25 AM   #1
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Math Focus: Number theory
Conservation of genus

If a genus-zero body is equivalent to two genus-zero bodies, what is a genus-one body equivalent to?
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July 5th, 2019, 01:03 PM   #2
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Genus-one plus k genus-zero, where k is any integer?
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July 6th, 2019, 09:50 AM   #3
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You may be right, mathman.

My initial question derives from the Banach–Tarski paradox.

How does B-T treat an object of nonzero genus?
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July 6th, 2019, 09:59 AM   #4
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I don't see how genus is at all relevant to the banach-tarski paradox.
In fact, the strong form of the paradox states that any compact subset of $\mathbb{R}^3$ can be decomposed into finitely many pieces. These pieces can be rotated and translated and be put together into any other compact subset of $\mathbb{R}^3$.
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July 6th, 2019, 01:08 PM   #5
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The key to the Banach-Tarski paradox is that the pieces are non-measurable sets.
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