Our recreational math geek lunch group got stuck on a question we need help to understand. I apologize in advance, if my explanation is not perfectly rigorous, as we are not professional mathematicians.
This is the question: What is the topological genus of 3d space if we remove a solid 3d ball from it?
We started from this example: A flat 2d plane minus a disk, the genus appears to be one, using both the cutting test (one cut to the hold doesn't create two pieces), and the rubber band test (a shrinking a loop around the hole doesn't reduce a point).
What are the genus tests for dimensions higher than a 2d plane?