TITLE: Meaning of "holes" counted by homology groups
QUESTION [10 upvotes]: In a lot of more or less informal introductions to simplicial homology
often the groups $H_k(X)$ of a topological space or CW space
are introduced as groups which "counting $k$-dimensional holes". I know that
is of course motivated by rather elementary examples but nevertheless
even if we discuss simple examples like sphere or torus it is
not clear to me what is preciesely meant by a "$k$-dimensional hole".
Can it be clarified? Note, that it's only about intuition, I know that all this 'hole counting approach' of homology groups cannot be formally approached, but even from 'informal' point of view I see several problems which I would like to clarify.
The two most common examples, the $2$-sphere $S^2$ and
the torus $T= S^1 \times S^1$ are discussed here in Wikipedia:  these two examples carry exactly the two properties of this 'hole terminology' which I find rather misleading or maybe just misunderstand.
The terminology of "holes" in case of $S^2$ looks to me intuitively rather acceptable up to the dimension choice, see further.
We call the $0$-holes the connected components. Since the sphere
is hollow, it is reasonable to say that is has a "hole".
But why this "hole" of $S^2$ is called $2$-dimensional hole?
Intuitively the hollow space inside of $S^2$ is $3$-dimensional,
therefore I not understand what is the logic behind the name
"$2$-hole" here.
Similary it is said that the circle $S^1$ has a $1$-dimensional hole.
But isn't this hole regarded from common sense $2$-dimensional?
Essentially this "hole" is the removed inner of a $2$-disc $D$ where
$S^1 = \partial D$. Can somebody clarify the 'logic' behind the
"dimension" of the holes in this setting.
Even more confusing is the notation of a hole for a torus $T$. According to
the 'logic' above a $k$-dimensional hole of a $k$-simensional
"surface" is the 'removed inner mass' which as observed in examples
before seemingly should be always contractible to a point.
But in case of torus the $1$-hole is not even contractible, since
it is homotopic to $S^1$.
That's confusing. Is it possible at least just for these two quite simple examples to precisely define what a $k$-hole is?

REPLY [2 votes]: The other answers are good at explaining how to think about the standard theory, but missing some discussion on why this formulation is preferable I think.
The punchline for your intuition: It is understandable to visualize topological spaces as subspaces of Euclidean space. But you should be aware where this breaks down. Why is the "hole" in a sphere not the void in the middle? Because if the world is a sphere and nothing else, the "void in the middle of a sphere" does not exist to discuss.
Say that we want to formulate a theory of holes in spaces. Before writing down any formal encoding, we need to be precise about what we mean by "hole", and it seems like there are two options to start us in that direction:

A hole is a void where material could have been.
A hole is a circle which is not filled with material.

These ideas are subtly different. To specify a hole in Version 1, we must say what material constitutes the void, and have a condition that allows us to say "that material is missing". Version 2 is the standard version. It specifies holes using circles, and everything we use to talk about holes in homology, including their dimension, is predicated on this.
To be more precise for Version 1, this approach may drive us to say: "A topological space $Y$ has a $k$-dimensional hole if $Y\subseteq X$ and there is a $k$-dimensional $U \subseteq X - Y$ with $\partial U \subseteq Y$." This was not an unreasonable starting point, but thinking carefully about it there is not clearly a workable theory to be had. Even if there were, the theory is "extrinsic": To specify a hole in $Y$, we must embed it in a bigger space $X$. By referencing a bigger space, we have made our job formalizing this theory a lot harder. Does our homology theory depend on the embedding? We no longer have $H_k(Y)$, we have $H_k(Y\subseteq X)$.
In version 2 (the standard approach), to specify a hole we provide a "witness": A circle in the space which is not filled. This theory is "intrinsic": A hole in $Y$ can be specified using only $Y$ without reference to anything else. A topological space $Y$ has a k-dimensional hole if there is a k-dimensional circle (better: a k-dimensional cycle) in $Y$ that lacks material in $Y$ filling it.