I told some advanced high-school students informally about homology (i.e. the "number of holes" in a space, by which I really mean the first Betti number), and one of them seemed to think that the homology of a $2$-sphere was $-1$. Their justification was that if you "add a hole" then you get a disc, which has zero holes, and therefore the original sphere must have one less than zero holes.
This student says that they have heard this somewhere, so I imagine they are referring to some related or historical construction. Does anyone know what they were likely referring to?
If you want to think of both steps in $3$-disk $\leadsto$ $2$-sphere $\leadsto$ $2$-disk as adding holes then you can only keep invariant the parity of the number of holes. In that case, Euler characteristic mod $2$ (which of course is the same as sum of Betti numbers mod $2$) is what you're looking for. If you are happy with saying that "an odd-dimensional hole cancels out an even-dimensional hole" then the Euler characteristic is a refinement. You can take negative reduced (as in reduced homology) Euler characteristic if you want it to match the number for the $2$-sphere and disk with finitely many punctures.