July 5th, 2019, 12:25 AM  #1 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 435 Thanks: 28 Math Focus: Number theory  Conservation of genus
If a genuszero body is equivalent to two genuszero bodies, what is a genusone body equivalent to?

July 5th, 2019, 01:03 PM  #2 
Global Moderator Joined: May 2007 Posts: 6,786 Thanks: 708 
Genusone plus k genuszero, where k is any integer?

July 6th, 2019, 09:50 AM  #3 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 435 Thanks: 28 Math Focus: Number theory 
You may be right, mathman. My initial question derives from the Banachâ€“Tarski paradox. How does BT treat an object of nonzero genus? 
July 6th, 2019, 09:59 AM  #4 
Senior Member Joined: Oct 2009 Posts: 841 Thanks: 323 
I don't see how genus is at all relevant to the banachtarski 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 $\mathbb{R}^3$. 
July 6th, 2019, 01:08 PM  #5 
Global Moderator Joined: May 2007 Posts: 6,786 Thanks: 708 
The key to the BanachTarski paradox is that the pieces are nonmeasurable sets.


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