Intuition of the meaning of homology groups

I am studying homology groups and I am looking to try and develop, if possible, a little more intuition about what they actually mean. I've only been studying homology for a short while, so if possible I would prefer it if this could be kept relatively simple, but I imagine it is entirely possible there is no real answer to my query anyway.

As I said above, I want to gain a little deeper understanding of what the n-th homology group actually means: I can happily calculate away using Mayer-Vietoris but it doesn't really give me a great deal of intuition about what the n-th homology group actually means. For example, with homotopy groups, the fundamental group is in some sense a description of how loops behave on the object in question, and it is obvious to me why that is what it is for say, the torus or the circle. However, I have no idea what, if anything, I am actually saying about a triangulable object when I talk about it having 0-th homology group this or 1st homology group that.

The best I have been able to find online or in my limited book selection is the brief description "intuitively, the zeroth homology group counts how many disjoint pieces make up the shape and gives that many copies of $\Bbb Z$, while the other homology groups count different types of holes". What 'different types of holes' are there, roughly speaking? I appreciate that it may often be completely non-obvious what the low-order homology groups are for some complicated construction, but perhaps in simpler examples it might be more explicable. Are there (simple) cases where I could say, just from looking something like e.g. the torus, what its zero-th or first or second etc. homology group was based on the nature of the object? I guess in the zero-th case it is, as my source (http://teamikaria.com/hddb/wiki/Homology_groups) above says, related to the number of disjoint pieces. Can we delve deeper than this for the other homology groups?

Any book/website suggestions would be welcomed (preferably websites as I am nowhere near a library!) - I have Hatcher but not a great deal else, and I haven't gleaned as much as I wish to from that alone. Of course I know that there is a great deal we don't know about homology groups even today, so I don't expect some magical all-encompassing answer, but any thoughts you could provide would be appreciated. I hope this question is appropriate for SE Mathematics, apologies if not! -M

Solution 1:

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.”

Solution 2:

"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 $H_{n-1}$.