What is the formal definition of a hole?

Question

In topology, there is a definition of "number of holes" of a manifold, like a torus. However, I have never seen the definition of hole by itself. Intuitively, a hole is a region of space where the manifold doesn't exist, but this is merely a necessary condition to be a hole, not a sufficient one. My question is, has someone given a formal and rigorous definition of a hole?

Answer 1

There are two ways that holes are measured precisely (that I know of). These are homotopy groups and homology groups.

I will describe the idea of 'one-dimensional holes'. These generalise to '$n$-dimensional holes' also. Consider a closed loop in your manifold $X$, which I will express as a map $\sigma:S^1\to X$. Such a loop may enclose a hole, e.g. in your example of a torus, $X=S^1\times S^1$, the map $\sigma:x\mapsto (x,a)$, for fixed $a\in S^1$, encloses a hole. However, there are also loops which do not enclose holes, e.g. any small enough circle on the torus. Homology and Homotopy have two different ways to tell apart the loops which detect holes and those that don't.

1. In homology groups, we ask whether it is possible to fill in the interior of the loop to get a disk whose boundary is the loop. If not, then you have a hole. In this context, the loops that can't be filled in are called cycles.

2. In homotopy groups, we ask if the loop can be continuously deformed to a trivial loop, one of the form $\sigma:x\mapsto a$ for a fixed $a\in X$, i.e. a constant map.

Actually fleshing this out further (e.g. What is the group structure on these groups? How do they work in dimension $n>1$) would be the subject of a book, such as Hatcher's Algebraic Topology, not a short answer like this.

Answer 2

To address your title question: There is no formal definition of a hole.

The purpose of the whole hole thing is to use our perception of familiar examples (annulus, torus) together with plain language (hole) in order to motivate topological concepts (Betti number, homology).

Once your understanding of these topological concepts rises to a sufficient level, you perceive that they go far beyond our intuitive concept of a "hole", and furthermore that concept no longer brings anything to the table.

So, at that point, you drop it.