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I'm still just a beginner in algebraic topology, but there's a specific problem I'd like to understand, which is how to classify maps from one space into another up to homotopy -- for instance, I've really enjoyed learning about the Pontryagin-Thom construction which yields homotopy classification of maps into S^2. For some applications that I'm interested in, it turns out that the homotopy classification of maps from manifolds into homogeneous spaces (of strictly lower dimension, if that helps) are of interest.

I guess what I'm asking is for a pointer in the right direction, since algebraic topology is such a large subject. I've read scattered results here and there on specific examples of the above, but I haven't found any systematic way of thinking about it yet. Is there one? Someone once said "equivariant cohomology" to me, is that useful?

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5 Answers 5

I'm pretty sure that what you're asking for is incredibly hard. In particular, homotopy groups of spheres (homotopy classes of maps from Sn to Sk) are not yet understood.

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Well I certainly expected that for the most general case. Are there any special cases of the homotopy classification problem (if the target space has low codimension, maybe?) which are understood? –  j.c. Oct 27 '09 at 2:19
    
Or is there any way to relate homotopy groups of spheres to maps between other spaces in an organized fashion? I guess the Whitehead theorem gives some information along the lines of necessary conditions for maps to be homotopic... –  j.c. Oct 27 '09 at 2:22
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Check out the Adams spectral sequence, which computes (stable) homotopy classes of maps from one space to another (one prime at a time).

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The adams spectral sequence certainly does compute the desired result. But it sounds like you are looking for something to do/understand. by homogeneous spaces do you mean G/H for two lie groups? in this case it is certainly possible that equivariant invariants would be important to look at.

if you mean things like BO(n) or BU(n) then these are things that we certainly have computations of the homotopy classes of maps into such spaces (which have models as homogeneous spaces in the above mentioned sense.

so if you could give me some more details about the scattered results you mentioned and what you mean by homogeneous space then i will edit this answer.

also to reference your comment on anton's answer, yes there are various filtrations of the homtopy groups of spheres that "we" understand. All these filtrations end up giving different versions of an adams spectral sequence that converge to a certain amount of the $\pi_n^S(S^0)$. the chromatic picture is the most interesting one that comes to my mind. to read a bit about this see tyler lawson's abelian varieties in homotopy theory paper, and perhaps go through his bibliography to find other sources.

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Homogeneous spaces tend to be buildable from spheres in finitely many steps by extensions by fibrations. If you pretend you understand $[X, S^n]$ for all $n$, then you can try to analyze the exact sequences that result from your fibrations.

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Very late, I know, but I just stumbled upon this question and I thought I would point out the reference http://arxiv.org/abs/0808.0024, which gives a nice modern discussion of this problem in the case when the homogeneous space is simply-connected and the space mapping in is $3$-dimensional. (This case was solved by Postnikov, in one of the early triumphs of obstruction theory).

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