Why do the homology groups capture holes in a space better than the homotopy groups?

Question

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.

Answer 1

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.

Answer 2

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.)

Answer 3

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.