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?