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Let $H_i$ be the $i$-th homology group of a topological space $X$ ($X$ to be read from context). The Wikipedia page on homology says the circle $S_1$ has $H_1$ equal to $\mathbb{Z}$, so that it "has" a "one-dimensional hole". Furthermore, $S_2$ has $H_2$ equal to $\mathbb{Z}$ because it has a two-dimensional hole, and $S_n$ has $H_n$ equal to $\mathbb{Z}$ because it has an n-dimensional hole.

I'm trying to come up with a naive interpretation of what it means to "have" an $n$-dimensional hole. $S_1$, embedded on the plane, encloses (ie. is the boundary of) a $2$-ball (aka. disk), $S_2$ encloses an $3$-ball, etc., so my naive interpretation would be, at any rate, that $S_n$ "has" an $(n+1)$-dimensional hole (which is false).

I'm sure this question can be answered with a little understanding of elementary homological machinery, but, for now: what is an intuitive explanation of an $n$-dimensional manifold "having" a $k$-dimensional hole?

(An $i$-dimensional "hole" --aka. a homology class-- seems to (somehow) be an element of $H_i:=\text{Ker }d_i /\text{Img }d_{i+1}$, which I think measures how much the underlying chain complex fails to be an exact sequence, since chain complexes satisfy $\text{Img }d_{i+1} \subseteq \text{Ker }d_i$ whereas an exact sequence would have equality.)

(This question doesn't provide much information.)

étale-cohomology
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1 Answers1

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I would not say that $S^1$ "encloses" a 1-dimensional hole, but rather it "has" a 1-dimensional hole. Certainly homologies measure something which seems to be related to what we intuitively think as "holes". However, it is hard to actually define "holes" and how to count them. As I understand, homologies provide a way to mathematically "count" them. Probably you may be tempted to embed a topological space into a Euclidean space and see the number of connected regions enclosed by it, but remember, "topology" of a topological space is something intrinsic to the space itself, not a specific embedding of a space.

In the spirit of this "intrinsic" topology, I think we should always try to think "in space", not "outside of the space". For instance, in case of $S^2$, imagine ourselves "living on $S^2$". Then what are holes? We cannot see $S^2$ from the outside. So our intuitive understanding of holes (such as enclosing a ball) cannot be applied to the creatures living inside the space. But still, homology provides a good way to measure a sort of topological-nontrivialness of the space from the insider's view.

The way I intuitively understand homologies is this: if $X$ has a non-vanishing $H_n(X;\mathbb{R})$, then it contains an $n$-dimensional closed oriented submanifold which cannot be shrinked to a point via "homological deformation". Here homological deformation means a deformation like homotopy, but which allows a manifold to "cancel out" when two differently oriented pieces meet. For example, in $\mathbb{R}^3$, a torus is homologically the same as the sphere, because we can shrink a meridian of a torus to actually "cut" a handle. Such transformation is homological, but not homotopical. (This is the difference between the homology and homotopy.)

In conclusion, I think we must think of homology groups (and homotopy groups) as some ways to define and count "the number of $n$-dimensional holes" of a topological space.

Henry
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