10

Greg Muller, in a post called Rational Homotopy Theory on the blog "The Everything Seminar" wrote

"I tend to think of homotopy theory a little bit like ‘The One That Got Away’ from mathematics as a whole. Its full of wistful fantasies about how awesome it would have been if things could only have worked out. Imagine if homotopy groups of spaces and homotopy classes of maps were as easy to compute as homology groups…"

What are the wonderful consequences that he is referring to?

Yemon Choi
  • 25,496
teil
  • 4,261
  • 3
    You could ask him by email, no? – Mariano Suárez-Álvarez Apr 18 '10 at 00:20
  • 8
    I'm a little suspicious of this sort of counterfactual. Wouldn't number theory be awesome if 37 were composite? – S. Carnahan Apr 18 '10 at 01:02
  • 6
    Scott: Z[sqrt(37)] has unique factorization like Z, but an infinite unit group unlike Z, so number theory can be more interesting when 37 is composite. Seriously, I wonder why this question is getting voted down. Anticipating that certain things might be feasible if you could do something else isn't that unreasonable. – KConrad Apr 18 '10 at 01:09
  • 6
    @Scott: 37 will always be prime, but it may be that someone in the future will find a better way to calculate homotopy groups. I see it as being similar to L-functions, no-one can determine their roots, but people are excited about L-functions because if their roots could be found then all sorts of excellent things would follow. But I don't know what excellent things would follow from improvements to homotopy theory. – teil Apr 18 '10 at 01:19
  • @ Mariano: I hadn't thought of that. Assuming that I could obtain his email address, do you think academics would be interested in answering emails from amateurs such as myself? – teil Apr 18 '10 at 01:24
  • 2
    I disagree with MSA that you should e-mail Greg, only because I am always in favor of this type of conversation being held in public. But commenting on his blog post seems to assuage my concern, and may get a more immediate response than posting here. – Theo Johnson-Freyd Apr 18 '10 at 03:22
  • Okay, those are pretty convincing arguments. I think something about the wording of the question set me off. – S. Carnahan Apr 18 '10 at 03:48
  • 6
    I'm not confident enough in my understanding of this stuff to post this as an actual answer, but one reason might be the Pontrjagin-Thom construction which yields an isomorphism between (framed, I think) cobordism groups and stable homotopy groups of spheres. As I understand it this machinery was developed in order to compute homotopy groups using manifold theory, but later with the advent of spectral sequences and other revolutions in homotopy theory the result became a useful way to prove theorems about manifolds. Still, if homotopy theory were easier then so would be manifold theory. – Paul Siegel Apr 18 '10 at 05:11
  • @Roland: In your response to Scott, you mention there might be better ways to compute the homotopy groups of spheres. Perhaps your question could be better shaped this (admittedly more narrow) way: "What is the computational complexity of determining the group structure for $\pi_n S^k$ ?" – Ryan Budney Apr 18 '10 at 05:43
  • @Ryan: As I understand it, he is not referring to something like: imagine a world where we knew the fundamental theorem of calculus, but we didn't know how to find anti-derivatives. Homotopies corresponds with anti-derivatives, but I don't know what corresponds to the fundamental theorem of calculus. – teil Apr 18 '10 at 11:15
  • edit: should be "As I understand it, he means something like: ..." – teil Apr 18 '10 at 11:32
  • Are you kidding me? Closed because it is not a real question? – teil Apr 18 '10 at 23:51
  • @Desolate: as far as I understand it, "not a real question" does not mean that it isn't a worthwhile question per se, but that in some people's judgment the question is more vague or more broad than is useful for MO. In particular, since people here have suggested that you could ask Greg Muller himself what he meant -- and I'm sure he would be happy to expand further on his remarks -- then perhaps other people feel there is no need for the question to remain open. Personally, I think your question is no worse than many other "big picture" ones. – Yemon Choi Apr 19 '10 at 03:52
  • 3
    Casting vote to reopen. See my previous comment. – Yemon Choi Apr 19 '10 at 03:52
  • 3
    depending on the generating hypothesis, we might not know that much. It seems like there is interplay between the use of an invariant and how easy it is to compute. for one thing, if homotopy were as easy as homology several people would be out of jobs. i think Siegel's comment is an excellent example, and one that probably has analogs in different fields. So a possible rephrasing of your question could be: Suppose we learned all of the (un)stable homotopy groups of spheres tomorrow, how would this affect people not strictly involved in homotopy theory? – Sean Tilson Apr 19 '10 at 04:22
  • 1
    ...

    and a philosophical answer to your original question could be that homotopy groups aren't the right thing to be looking at.

     – Sean Tilson Apr 19 '10 at 04:22
  • BTW: do either of the answers below go some way to answering your question? are you minded to accept either of them? If not, then I'm sure both the respondents would appreciate a comment – Yemon Choi Apr 25 '10 at 09:34
  • @Yemon: Several answers can provide insight, I'm uncomfortable with singling one out and labeling it as the accepted answer.

    These 2 answers are not what I had in mind, but I don't have enough understanding to comment on them.

     – teil Apr 26 '10 at 00:16

2 Answers2

13

Those who reply that being able to compute homotopy groups is akin to being able to compute values of L-functions are perhaps unwittingly referring to a great parallel: stable homotopy groups of spheres are the "algebraic K-theory of finite sets" while, while the algebraic K-theory of number fields is conjecturally related to special values of L-functions (the Quillen-Lichtenbaum conjecture).

What Muller is almost certainly referring to is that homotopy is a powerful functor. For example, Freyd's generating hypothesis conjecture implies that the stable homotopy classes of maps between two finite CW complexes can be faithfully viewed through the module maps, over stable homotopy of spheres, of their stable homotopy.

Thus, if we were really good at understanding homotopy groups, then presumably we'd be that much closer to understanding various questions which can be reduced to or require significant input from homotopy theory. Classically, such applications included classifying manifolds up to diffeomorphism - Kervaire and Milnor reduced this question for spheres to one in homotopy. Because rational homotopy groups - really, rational homtopy theory - are more approachable (for example, one gets an upper bound on the homotopy groups of a simply-connected X through the Harrison homology of its cohomology ring), they have been widely-applied in geometry. See for example the book "Algebraic Models in Geometry" by Felix-Oprea-Tanre, which discusses topics ranging from group actions to symplectic manifolds to geodesics to toric topology to configuration spaces. One could presumably get much sharper results in all of these areas with homotopy over the integers.

But homotopy theorests are generally more excited these days by potential application to algebra itself. The sphere spectrum, whose coefficients are the stable homotopy groups of spheres, plays the same role in the world of $E_\infty$-algebra ("derived" commutative algebra) as the integers do in algebra - this is one philosophical starting point of Lurie's opus.

Dev Sinha
  • 4,960
12

Homology groups and homotopy groups are two sides of the same story. Homotopy groups tell us all the ways we can have a map Sn → X, and in particular describe all ways we can attach a new cell to our space. On the other side, the homology groups of a space change in a very understandable way each time we attach a new cell, and so they tell us all the ways that we could build a homotopy-equivalent CW-complex. In cases where we can understand both of them, we can get things like complete theorems about classification of spaces.

Here's an example where we can compute: classification of the homotopy types of compact, orientable, simply-connected 4-manifolds. (I originally saw this is Neil Strickland's bestiary of topological spaces.)

Poincare duality tells us that the homology groups are finitely generated free in degree 2, ℤ in degrees 0 and 4, and zero elsewhere. We can cut out a closed ball, and get an expression of the manifold as obtained from a manifold-with-boundary N by attaching a 4-cell. The Hurewicz theorem tells us that we can construct a map from a wedge of copies of S2 to N which induces an isomorphism on H*, and by the (homology) Whitehead theorem this is a homotopy equivalence. So our original manifold is obtained, up to homotopy equivalence, by attaching a 4-cell to $\bigvee S^2$.

How many ways are there to do this? It is governed by $\pi_3 (\bigvee S^2)$, which we can compute because it's low down enough. This homotopy group is naturally identified with the set of symmetric bilinear pairings $H^2(\bigvee S^2) \to \mathbb{Z}$, and this identification is given by seeing how the cup product acts after you attach a cell. So these 4-manifolds are classified up to homotopy equivalence by the nondegenerate symmetric bilinear pairing in their middle-dimensional cohomology.

Some of what we used here is general and well-understood machinery about homology, homotopy, and their relationships. Wouldn't it be nice if the standard tools were always so effective? But the real meat is that we have a complete understanding of homology and homotopy in the relevant ranges. It turns our questions about classification into questions about pure algebra. For questions that require specific knowledge about higher homotopy groups of spheres (or even lower homotopy groups of complicated spaces), it is much harder to get answers. There aren't a lot of spaces where we have complete understanding of both the homology groups and the homotopy. We have tools for reconstructing the former from the latter but their effectiveness wears down the farther out you try to go.

There are categories that are somewhat like the homotopy category of spaces where we can get an immediate and specific understanding of both sides of the coin.

One such example is the category of chain complexes over a ring R. There, our fundamental building block is R itself. The homology of any chain complex tells us both how R can be mapped in modulo chain homotopy, and how complicated any construction of the underlying chain complex must be. A more complicated example would be the category of differential graded modules over a DGA, where the divide between how things can be constructed and how things can have new cells attached is, at the very least, governed by the complexity of H* A as a ring, and then by the secondary algebraic operations if A is far from being anything like formal.

Another such example is the rational homotopy theory of simply-connected spaces you mentioned. There, homology and homotopy are roughly something like the difference between a ring's underlying abelian group structure and how you build it using generators and relations.

So you might think of the complexity of homotopy groups as telling us how much more complicated spaces are than chain complexes.

Tyler Lawson
  • 51,259