What Does It Mean to Say that Betti Numbers Count Holes?

Question

There seem to be two distinct notions of "hole" in homology. The first is cycles mod boundaries, which I'm fairly comfortable with. The second is Betti numbers, which are more confusing to me. The first issue: Betti numbers for singular homology think a non-empty space has an extra hole. Oh, well. The other issue is more serious. If the homology groups of a space have no torsion, then I'm fine with Betti numbers. But, if the Betti numbers of a space count the number of holes in it, then that would mean that the real projetive plane has no holes it is not contractible. The idea behind Betti numbers seems to be reducing the amount of clutter in homology groups. For example, if your homology group is $\mathbb{Z}^4$, then the corresponding Betti number is $4$, because all the other cycles can be built out of the four generators of the homology group. But why do cycles that are not boundaries but belong to the torsion suddenly stop counting as holes here? Like, the real projective plane has a hole.

Answer 1

This is obviously informal, but I would propose that Betti numbers try to count 'true' holes. I consider the $\Bbb Z/2\Bbb Z$ torsion in the first homology of the projective plane as indicating a nontrivial twisting, a loop that cannot be unwound - in particular, an obstruction against being contracted - but it is not a true hole because it does not persist; you can overcome this obstruction by merely walking around it a second time, twisting it over itself so that it is now untwisted. However the hole in $S^1$ cannot be beat; it is persistent and not just due to some kind of twisting (which can be untwisted), no amount of struggling back and forth can get the hole to disappear. $\Bbb Z$-summands of homology tend to more closely match our own intuitive perception of hole, and torsion summands - though significant and certainly carrying valuable information - are less hole-like and are more twist-like, in my view. If you visualise partial attempts to construct the real projective plane, this is emphasised: you're furiously trying to squish down a copy of $D^2$ onto a copy of $S^1$ with an antipodal identification on the boundary and you're manically twisting bits and pieces over and under each other... but you're not exactly boring any holes.

Thus, by killing torsion, not only do you have an algebraically far simpler and digestible object (indeed if you look at $H_\ast(-;\Bbb Q)$ you get a vector space) but what you get fits the intuitive description better. Consider also that, historically, Betti numbers came first and homology came after, and a first attempt at classifying holes would surely focus on the 'true' holes which resemble what we see in reality.