Now following a (probably) super soft question: Homotopy theory seems to be one of the most important current reseach topics in topology. While (co-)homology seems to be quite easy (or at least not too hard), homotopy theory seems so be quite complicated. What is the reason for this?

Please note, that the question is not why homotopy is hard to calculate. I know about the failure of excision for higher homotopy groups, but this doesn't explain why for example the homotopy-groups of spheres are so complicated. It could be the case, that we have to put in great effort to find, that for example $\pi_n(\mathbb{S}^k)=H_n(\mathbb{S}^k)$ (wether by Hurewitz or not). But it turns out that those groups are not just very hard to compute, but there also seems to be hardly any pattern.

So what is the conceptual difference between homotopy and homology making the one "easy" and the other one "hard"?

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    $\begingroup$ Very broadly, I think problems start to arise when we relax our notion of equality. Since homotopy is a very loose notion of "equality" between maps, there's a lot of flexibility in the kinds of objects we're looking at $\endgroup$ – leibnewtz Mar 31 '18 at 0:59
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    $\begingroup$ I think your question is a bit backwards: most things are hard. The surprising thing is that homology is easy. $\endgroup$ – Eric Wofsey Mar 31 '18 at 1:05
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    $\begingroup$ "Please note, that the question is not why homotopy is hard to calculate." But that's why homotopy is complicated. $\endgroup$ – Randall Mar 31 '18 at 3:20

If you see homotopy groups as mere abelian groups, then it is not very surprising that you're not seeing any pattern at all. First of all, there's substitution $\pi_k(S^l) \times \pi_l(S^h) \to \pi_k(S^h)$, which is linear on left but not so on right; for example, precomposing with stable Hopf map $S^{k+1} \to S^k$ is quadratic. Then there's Whitehead bracket, making all homotopy groups of fixed space into graded superLie ring. (Ingeniously) using this structure and some very basic homotopy theory, it's possible to calculate 10th stem (i. e. $\pi_{n+10}(S^n)$) as it's done in "Composition Methods in Homotopy Groups of Spheres" by Hiroshi Toda. I'd say it is not surprising that big collection of graded Lie algebras in some complicated way acting on each other show some interesting behaviour — and what's most intriguing, it's the most natural algebraic structure coming directly from algebraic topology.

Another point is that homotopy groups are everywhere — homology is homotopy of infinite symmetric power by Dold-Thom theorem, algebraic K-theory is homotopy, so you couldn't expect it to be something simple.

Third aspect is that we usually think of spaces in cellular model, that is iterated cones of maps of spheres of different dimensions. But there's also a cocellular model — Postnikov tower, which is fibration tower with $K(\pi_k, k)$ as fibers. From this point of view computation of homotopy is very easy — but simple objects are very different.


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