Exact meaning of homology

Question

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Soft Question - Intuition of the meaning of homology groups

I've been studying some homology recently and I know it supposedly counts $n$-dimensional holes. For example, the torus has first homology group $H_1(T^2) = \mathbb{Z} \times \mathbb{Z}$ where one copy of $\mathbb{Z}$ corresponds to the middle hole of the tire and the second copy of $\mathbb{Z}$ corresponds to its hollow inside. $H_2(T^2) = \mathbb{Z}$ where $\mathbb{Z}$ corresponds to the middle hole but considered in $2$ dimensions, I think.

Is that correct? Or what does a $2$-dimensional hole look like?

Then looking at the Klein bottle gives $H_1(K) = \mathbb{Z} \times \mathbb{Z}_2$. This is where my already shallow understanding ends: if there is a copy of $\mathbb{Z}_{n}$ in the homology, what sort of hole is there in the space?

Many thanks for your help.

Answer 1

"it counts n-dimensional holes" is simply a slogan, but reality is more complicated than that... In particular, your title question, «what is the exact meaning of homology?», is more or less unanswerable in the sense you want.

I think you should try to consider as many examples as you can, so as to build an intuition about what homology actually counts.

For example, consider the real projective plane $\mathbb P^2(\mathbb R)$. Do you know its fundamental group? Do you see why it is $\mathbb Z_2$. Do you know its $H_1$? Can you find an explicit generator $c$ for this latter group? Can you see why $2c=0$? Can you find a two cycle whose boundary is $2c$?

The torus $T=S^1\times S^1$ has two $1$-dimensional holes. Can you see them? How can you detect them concretely? How can you tell them apart?

When you have exhausted $\mathbb P^2(\mathbb R)$ and $T$, pass on to other spaces. There are many!

 

Also, you ask what a $2$-dimensional hole looks like... Well, do you know what a $1$-dimensional hole looks like? How can you detect a $1$-dimensional hole?

For example, if $L$ is a line in $\mathbb R^3$, then the set $X=\mathbb R^3\setminus L$ has a $1$-dimensional hole. You can detect it in that there are certain vector fields on $X$ which are not the gradient fields of any function $X\to\mathbb R$. One way to see this is to find an example of such a field $F$ such that the circulation of $F$ along a closed curve is not zero.

Now let $P$ be the origin in $\mathbb R^3$. There is no $1$-dimensional hole in $Y=\mathbb R^3\setminus\{P\}$. Can you see why? (as opposed to computing $H_1(Y)$) Also, $Y$ has a $2$-dimensional hole: can you see it? how can you detect its presence?