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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?

user107952
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2 Answers2

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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.

Joshua Tilley
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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.

Lee Mosher
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    Interestingly, it seems like this phenomenon appears in a number of places in mathematics. For instance, isomorphisms are often introduced to students as a way of saying that two algebraic structures are "the same", but ultimately, we develop an intuition for what isomorphisms are, and the notion of "sameness" ends up a feeling a little loose. After all, there are many other notions of "sameness", such as equality, elementary equivalence, canonical identification, etc. – Joe Jan 19 '24 at 22:24
  • @Joe This is also the case in physics where one starts with tangible phenomena (mechanics, thermodynamics to some extent) and then goes more and more into abstract concepts that come naturally. You end up "trusting the math" and not attempting to imagine the unimaginable. A typical example is the progression of visualization on 1D → 2D → 3D and then there is a split: the handwaving physics try to build a mental model of a 4D while the ones who work in the field do not need to. – WoJ Jan 21 '24 at 09:49
  • @Joe (cont'd) To be clear: I am a huge proponent of "handwave physics" as it brings this beautiful subject to more people, it is just that it breaks at some point and it does not make much sense to try to squeeze more of it. – WoJ Jan 21 '24 at 09:50