Intuition of the meaning of homology groups

Let's restrict ourselves to orientable spaces that are homotopic to CW complexes. In low dimensions, there is a very intuitive way to think of homology groups. Basically, the rank of the $n$-th dimensional homology group is the number of $n$-dimensional “holes” the space has. As you stated in your example, for $H_0$, this is counting connected components. Moving to $H_1$, we are counting literal holes. The torus has $H_1\cong \mathbb Z \oplus \mathbb Z$ since it has two holes, one inside and one outside.

You can think of a 2-dimensional hole as an empty volume. The best analogy I’ve heard is to think of your space as an inflatable object. The rank of the second homology group is the number of different plugs you’d need to blow air into to inflate it. The torus has one empty volume, so you’d only need one plug to inflate it. If you take the wedge of two 2-spheres, you’d need two different plug to inflate it, one for each empty volume, so it has rank 2.

As is usual in topology, we now wave our hands and say “it works the same for higher dimensions.”

"Cycles modulo boundaries" can get you surprisingly far. Recall that for a simplicial complex $X$, the $n^{th}$ homology $H_n(X)$ is $Z_n/B_n$, where $Z_n = \ker(d_n : C_n \to C_{n-1})$ is the group of cycles and $B_n = \text{im}(d_{n+1} : C_{n+1} \to C_n)$ is the group of boundaries. Some low-dimensional cases:

• When $n = 0$, a cycle is a linear combination of $0$-simplices in $X$, and a boundary is a linear combination of $0$-simplices lying in the same connected component of $X$ such that the sum of their coefficients is zero. So $Z_0/B_0$ is precisely the free abelian group on the connected components of $X$.
• When $n = 1$, a cycle is exactly what it sounds like: a linear combination of cycles in $X$ (closed paths made out of $1$-simplices). A boundary is also exactly what it sounds like: a linear combination of cycles in $X$ that bound $2$-simplices. So $Z_1/B_1$ describes the failure of $1$-cycles in $X$ to bound $2$-simplices (which is the precise sense in which it measures "$1$-dimensional holes").

For $n \ge 2$ I have trouble concisely describing what a cycle is beyond "a linear combination of $n$-simplices with zero boundary." I believe this is roughly like a linear combination of collections of $n$-simplices which together form an $n$-sphere in $X$, at least for sufficiently nice triangulations. A boundary is a linear combination of such things which bound $n+1$-simplices. So again $Z_n/B_n$ measures the failure of $n$-cycles to bound $n+1$-simplices, which is the precise sense in which it measures "$n$-dimensional holes."

$H_0$ and $H_1$ are probably easier to identify than the others in general, since connected components are intuitive and $H_1$ is just the abelianization of the fundamental group. If $X$ is a connected $n$-manifold, then $H_n \cong \mathbb{Z}$ if $X$ is orientable and $0$ otherwise, the idea being that an $n$-cycle has to involve all of the $n$-simplices in $X$ appropriately oriented so that their boundaries cancel, and such a linear combination is unique up to scalar multiplication and equivalent to providing an orientation for $X$. And if $X$ is a compact orientable $n$-manifold, then Poincaré duality indirectly relates $H_1$ to $H_{n-1}$.

To a certain extent, there is no great answer that works in complete generality, or at least in a way where you can have a reasonable intuition about it, enough to see at a glance what the homology of high dimensional spaces will be. Part of this is because of the difficulty of picturing higher dimensional spaces. However, we can get a glimpse of what the homology records by considering some examples.

First, understanding the finer details of homology is hard, so we will look at a simpler invariant, the betti numbers $b_i=\dim_{\mathbb{Q}} H_i(X;\mathbb{Q})$. This disregards all sorts of subtle information which is safe to ignore for starting to build your intuition.

$b_0$ is the number of connected components. $H_1(X)$ is the abelianization of $\pi_1(X)$, and so $b_1$ is the number of "independent loops" in $X$, or the number of holes . The best example of this looking at surfaces. Take the $3$-dimensional sphere, and attach $g$ handles to it. Now, take the surface which is the boundary of this three dimensional object. For each handle, we have a loop that goes around the handle (in the way you might pick up a coffee mug), and a loop that goes from the body, into the handle and back again. For this, imagine that inside the $3$-dimensional space, you took a rubber band that went through the handle, and you shrunk the band until it was taut against the surface.

So we have $b_0=1$, $b_1=2g$, and $b_2=1$ (In general, with a compact orientable $n$-manifold, $b_n=1$). However, instead of viewing the surface as coming from attaching handles to the $3$-sphere, we can view it as punching $g$ holes through the sphere. The fact that $b_1=2g$ and not $g$ might be a little disconcerting, as we have $g$ holes. This is remedied somewhat by the fact that the handle-body (the object before we took the surface) is homotopy equivalent to a wedge of $g$ circles, and so it does have $b_1=g$. However, this shows how subtle it is to say that $b_1$ is "the number of holes", as it makes it difficult to say what exactly a hole is.

To get a grasp about what's going on in the other directly, let's consider $\mathbb{R}^n\setminus \{0\}$. This is putting a hole in the space. Let's try to detect the hole by using spheres. We can't encase the hole inside of a $S^k$ with $k<n-1$, because there is enough ambient space that we can move the sphere past the hole. However, we can encase it inside of an $n-1$ sphere. You can build some more intuition about detecting holes with spheres by looking at $\mathbb{R}^n\setminus \mathbb{R}^m$, and thinking about when we can encase the "hole" inside a sphere.

Unfortunately, this has a lot more to do with higher homotopy groups. There are relations between homology and homotopy groups, but the connection is subtle. For example, while the Hurewicz isomorphism theorem gives a close connection in one dimension (for any particular space), there is a Hurewicz map in all dimensions, but it is not in general easy to understand what it does.

One good way of understanding homology of CW complexes is with cellular homology. In general, this is still hard to understand, but if you have a buffer between the dimensions of the spheres used to construct the space (e.g. complex projective spaces, which are built up out of even-dimensional spheres), homology is counting the spheres used to construct the space.

Beyond these things, I can only recommend that you let your intuition be built up from examples. Sadly, there is no magic bullet that will give you a great understanding.