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This is a follow-up to another question.

A good interpretation of having an $n$-dimensional hole in a space $X$ is that some image of the sphere $\mathbb{S}^n$ in this space given by a mapping $f: \mathbb{S}^n \rightarrow X$ cannot shrink down to a point. The matter of "shrinking to a point" is best expressed by being $f$ homotopic to some constant map. Next, the homotopy groups $\pi_n$ can be defined as the homotopy classes of base-point preserving maps from $\mathbb{S}^n$ to $X$. In this way it might be argued that the homotopy groups $\pi_n$ should best capture the holes in $X$.

But this is not so. One has the most satisfying result that for $i \geq 1$ the homology $H_i (\mathbb{S}^n) $ is nontrivial iff $n = i$. But the higher homotopy groups of spheres are very complicated.

Why does this complication occur? Why are homology groups far better for capturing the holes than the homotopy groups, which are intuitively better suited, but are not actually so? In the case of $1$-dimensional holes, the homology $H_1$ and $\pi_1$ captures the holes equally well; but of course in this case the former is the abelianization of the latter.

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Akela
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  • I asked a second question on a different facet of the same matter, following the guidelines in the how to ask page: mathoverflow.net/howtoask#breakitdown – Akela May 14 '10 at 22:58
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    I don't like the use of the term "hole", but in any case, one could argue the opposite: The complexity of homotopy groups is a reflection of the complicated ways in which holes can be manifested. For example, a connected space with simple nonabelian fundamental group will have trivial first homology, but I would claim that it has a nontrivial one dimensional hole. – S. Carnahan May 15 '10 at 01:32
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    Think about the Hopf map from $S^3$ to $S^2$. This shows that a 3-dimensional sphere can wind around, in a non-trivial way, a 2-dimensional "hole" (using your terminology). So one reason that homotopy groups are complicated is that higher dimensional spheres can wind around lower dimensional holes. – Emerton May 15 '10 at 06:21
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    If "hole" is not the acceptable terminology, then please suggest a better one. I thought it best reflects the intuitive picture. – Akela May 15 '10 at 06:25
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    I think a better terminology would be "cycle." Then the answer is that homology is better at detecting cycles than homotopy is because cycles are defined to be what homology detects. (-:O – Mike Shulman May 15 '10 at 06:40
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    To elaborate on the Hopf map idea, the reason higher-dimensional things can "wrap around" lower-dimensional one (the only reason, over the rationals) is that preimages of subspaces of the lower-dimensional objects can be linked. – Dev Sinha May 15 '10 at 16:35
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    Like the others, I think it all depends on what you mean by "hole". Roughly: Disk shaped holes give you homotopy groups, simplicial complex shaped holes give you homology, manifold shaped holes give you some kind of cobordism. – Dylan Wilson Jul 03 '17 at 01:29
  • @DylanWilson -- that's nice, but not I think quite accurate. Homology captures holes that look like simplicial complexes which look like manifolds with codimension two singularities. – HJRW Jul 03 '17 at 08:34
  • @HJRW while I wasn't trying to be precise, I think that particular part of my statement is nevertheless true. If you build a bordism theory with maps from oriented simplicial complexes then you get back ordinary homology. It is also true that one need only add some restricted class of singularities to some bordism theory of pl-manifolds before every homology class is represented, as you mention, but that doesn't contradict what I said. – Dylan Wilson Jul 03 '17 at 13:44
  • @DylanWilson, I suppose that's true if you insist that your invariants are abelian. But the non abelian boundary of a simplicial complex looks more like homotopy than homology. – HJRW Jul 04 '17 at 03:48
  • @HJRW I don't think I understand what that means... "non-abelian boundary"? Also homotopy groups are abelian after the first one... – Dylan Wilson Jul 04 '17 at 03:54
  • @DylanWilson , the point is that homology isn't what happens when you insist on working with simplicial complexes; it's what happens when you insist on doing linear algebra. – HJRW Jul 04 '17 at 04:02
  • @HJRW Sorry, I still don't understand what you're saying, but maybe this isn't the best place to sort it out. – Dylan Wilson Jul 04 '17 at 04:06
  • @DylanWilson -- agreed. :) – HJRW Jul 04 '17 at 09:01

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Homology also has complicated and unintuitive aspects if one goes beyond nice spaces like CW complexes. A surprising example of this is the subspace of Euclidean 3-space consisting of the union of a countable number of 2-spheres with a single point in common and the diameters of the spheres approaching zero. (This is the 2-dimensional analog of the familiar "Hawaiian earring" space.) Then the amazing fact is that the n-th homology group of this space with rational coefficients is nonzero, and even uncountable, for each n > 1. This was shown by Barratt and Milnor in a 1962 paper in the AMS Proceedings.

The result holds also with integer coefficients, with homology classes that are in the image of the Hurewicz homomorphism. So one could say that this strange behavior comes from homotopy groups but just happens to persist in homology. I would guess that there are also examples where the homology is not in the image of the Hurewicz homomorphism, so it does not come directly from homotopy groups.

Allen Hatcher
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For another example that homology and homotopy capture holes differently: note that the curve $C$ in the bitorus below is homologous to zero but not homotopic to zero.

Also, a remarkable difference between them is that while an homology chain can be subdivided into smaller chains, a map from a sphere cannot be subdivided into smaller maps from a sphere. Historically, higher homotopy groups were only pursued after Hopf's fundamental example of a map $S^3 \to S^2$ not homotopic to zero, which was already commented above (I recommend the reading of H. Samelson "$\pi_3(S^2)$, H. Hopf, W. K. Clifford, F. Klein", In: History of Topology, p.575-578, Elsevier, Amsterdam, 1999.)

Lucas Seco
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    Much more generally, the Kan--Thurston theorem shows that there are finite cell complexes with fairly 'small' homotopy groups (zero in dimension >1) but arbitrarily complicated homology groups. – HJRW Jul 03 '17 at 08:35
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I am no specialist, but I believe that it is not that the homology and homotopy capture better or worse the holes. I believe that they capture them differently.

Obviously, it is far easier to compute homology groups, but on the other hand, they give much less information in a lot of cases. I like Whitehead Theorem ( http://en.wikipedia.org/wiki/Whitehead_theorem ) as an example of how powerfull can homotopy groups. I don't know a counterpart for homology groups.

The other example I like, is the two dimensional torus. On the one hand, the two dimensional homology is non trivial while the second homotopy group is trivial which could indicate that homology is capturing a hole that homotopy does not capture. However, homotopy has already captured the holes of the torus (since the first homotopy group is non trivial). I think this example shows how it should be difficult to define the "dimension" of a hole.

So, I believe that homotopy groups should be considered all together and they will escentially capture all the holes.

rpotrie
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    One could think of the torus a different way: you can wrap a two-dimensional cycle around it, namely iself; this gives the fundamental class in $H^2$. But this two-dimensional cycle is not a 2-sphere, it is a 2-torus, and so it can't be detected by $\pi_2$. – Emerton May 15 '10 at 14:19
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    Whitehead's theorem does have a homology counterpart (for simply connected spaces, say); see Hatcher's AT, Cor. 4.33. E.g. by Whitehead plus Hurewicz, a simply connected homology $n$-sphere is a homotopy $n$-sphere. – Tim Perutz May 15 '10 at 15:56