Since Ronnie Brown and his collaborators have come up with a general proof of the higher Van Kampen theorems, what impediments are there to using these to compute the unstable homotopy groups of spheres?
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asked Sep 24, 2010 at 4:05
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1$\begingroup$ Which result of Brown are you referring to? The higher Van Kampen theorem is usually called "Stover's spectral sequence" no? I haven't seen a computation of anything using any of these techniques, but I'm not sure which result of Brown you're referring to. $\endgroup$– Ryan BudneyCommented Sep 24, 2010 at 4:14
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4$\begingroup$ @Ryan.Have a look at Ronnie Brown's web pages for a history of the van Kampen theorem in its higher dimensional forms and an extensive bibiography including calculations.Yes,there are spectral sequences that give some (sometimes poor) information in van Kampen type situations, for instance a paper in Topology by Artin and Mazur as well as the one that you mention, but the original vKT allows the complete calculation of a homotopy type of a union from the parts. The first Brown-Higgins generalisation allowed calculations that were difficult to do otherwise, but this is low dim. homotopy. $\endgroup$– Tim PorterCommented Sep 24, 2010 at 6:56
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4$\begingroup$ I think that it's worth pointing out that algorithms existed long before Brown's work for computing homotopy groups of spheres. The problem is that those algorithms are too slow to be of much practical use. I expect that the same is true for any answer coming from some kind of "higher van Kampen" theorem. There are patterns in the homotopy groups of spheres (eg the chromatic picture of the stable stems), but they are too complicated to arise from any simple description. $\endgroup$– NikitaCommented Sep 24, 2010 at 13:53
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1$\begingroup$ @ Ryan You can get lost in Ronnie's website. A good place to start might be bangor.ac.uk/~mas010/hdaweb2.htm but there are links to articles and slides from that page. (He gave a talk on a related topic last week in Birmingham and the slides are there somewhere on his web pages.) $\endgroup$– Tim PorterCommented Sep 24, 2010 at 20:39
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1$\begingroup$ see mathoverflow.net/questions/22837/… for more. $\endgroup$– Tim PorterCommented Sep 24, 2010 at 20:47
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3 Answers
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Here are some answers on the HHSvKT - I have been persuaded by a referee that we ought also to honour Seifert.
These theorems are about homotopy invariants of structured spaces, more particularly filtered spaces or n-cubes of spaces. For example the first theorem of this type
Brown, R. and Higgins, P.~J. On the connection between the second relative homotopy groups of some related spaces. Proc. London Math. Soc. (3) \textbf{36}~(2) (1978) 193--212.
said that the fundamental crossed module functor from pairs of pointed spaces to crossed modules preserves certain colimits. This allows some calculations of homotopy 2-types and then you need further work to compute the 1st and 2nd homotopy group; of course these two homotopy groups are pale shadows of the 2-type.
As example calculations I mention
R. Brown ``Coproducts of crossed $P$-modules: applications to second homotopy groups and to the homology of groups'', {\em Topology} 23 (1984) 337-345.
(with C.D.WENSLEY), `Computation and homotopical applications of induced crossed modules', J. Symbolic Computation 35 (2003) 59-72.
In the second paper some computational group theory is used to compute the 2-type, and so 2nd homotopy groups as modules, for some mapping cones of maps $ Bf: BG \to BH$ where $f:G \to H$ is a morphism of groups.
For applications of the work with Loday I refer you for example to the bibliography on the nonabelian tensor product
http://groupoids.org.uk/nonabtens.html
which has 144 items (Dec. 2015: the topic has been taken up by group theorists, because of the relation to commutators) and also
Ellis, G.~J. and Mikhailov, R. A colimit of classifying spaces. {Advances in Math.} (2010) arXiv: [math.GR] 0804.3581v1 1--16.
So in the tensor product work, we determine $\pi_3 S(K(G,1))$ as the kernel of a morphism $\kappa: G \otimes G \to G$ (the commutator morphism!). In fact we compute the 3-type of $SK(G,1)$ so you can work out the Whitehead product $\pi_2 \times \pi_2 \to \pi_3$ and composition with the Hopf map $\pi_2 \to \pi_3$.
These theorems have connectivity conditions which means they are restricted in their applications, and don not solve all problems! There is still some interest in computing homotopy types of some complexes which cannot otherwise be computed. It is also of interest that the calculations are generally nonabelian ones.
So the aim is to make some aspects of higher homotopy theory more like the theory of the fundamental group(oid); this is why I coined the term `higher dimensional group theory' as indicating new structures underlying homotopy theory.
Even the 2-dimensional theorem on crossed modules seems little known or referred to! The proof is not so hard, but requires the notion of the homotopy double groupoid of a pair of pointed spaces. See also some recent presentations available on my preprint page.
Further comment: 11:12 24 Sept.
The HHSvKT's have two roles. One is to allow some calculations and understanding not previously possible. People concentrate on the homotopy groups of spheres but what about the homotopy types of more general complexes? One aim is to give another weapon in the armory of algebraic topology.
The crossed complex work applies nicely to filtered spaces. The new book (pdf downloadable) gives an account of algebraic topology on the border between homotopy and homology without using singular or simplicial homology, and allows for some calculations for example of homotopy classes of maps in the non simply connected case. It gets some homotopy groups as modules over the fundamental group.
I like the fact that the Relative Hurewicz Theorem is a consequence of a HHSvKT, and this suggested a triadic Hurewicz Theorem which is one consequence of the work with Loday. Another is determination of the critical group in the Barratt-Whitehead n-ad connectivity theorem - to get the result needs the apparatus of cat^n-groups and crossed n-cubes of groups (Ellis/Steiner).
The hope (expectation?) is also that these techniques will allow new developments in related fields - see for example work of Faria Martins and Picken in differential geometry.
Developments in algebraic topology have had over the decades wide implications, eventually in algebraic number theory. People could start by trying to understand and apply the 2-dim HHSvKT!
Edit Jan 11, 2014 Further to my last point, consider my answer on excision for relative $\pi_2$: https://math.stackexchange.com/questions/617018/failure-of-excision-for-pi-2/621723#621723
See also the relevance to the Blakers-Massey Theorem on this nlab link.
December 28, 2015 I mention also a presentation at CT2015 Aveiro on A philosophy of modelling and computing homotopy types. Note that homotopy groups are but a pale "shadow on a wall" of a homotopy type. Note also that homotopy groups are defined only for a space with base point, i.e. a space with some structure. My work with Higgins and with Loday involves spaces with much more structure; that with Loday involves $n$-cubes of pointed spaces. As with any method, it is important to be aware of what it does, and what it does not, do. One aspect is that the work with Loday deals with nonabelian algebraic models, and obtains, when it applies, precise colimit results in higher homotopy. One inspiration was a 1949 Theorem of JHC Whitehead in "Combinatorial Homotopy II" on free crossed modules.
answered Sep 24, 2010 at 15:21
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3$\begingroup$ Can't get any more definitive than this! $\endgroup$ Commented Sep 24, 2010 at 16:36
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7$\begingroup$ @ Harry. I thought that the best answer to your question was two fold. Firstly to e-mail Ronnie as he knows the stuff inside out and has lots of links to his pages available, so I did that. (You see the result. (We're quick off the mark in North Wales, :-))) The second stage should be to up-date the n-Lab entry and bring in some of the other stuff that RB did, e.g. the material with Loday. That will take time and hopefully others will contribute to it as well. $\endgroup$ Commented Sep 24, 2010 at 16:49
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$\begingroup$ I figured you had a hand in this! =), indeed! $\endgroup$ Commented Sep 24, 2010 at 16:55
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Ronnie and collaborators' HHvK theorems are essentially all for crossed complexes and similar. These are, as I'm sure you're aware, partially linearised homotopy types. In particular, for a simply connected space, they are just chain complexes. So for $S^k, k>1$ it's not more useful than the homology.
answered Sep 24, 2010 at 4:45
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6$\begingroup$ @David You are forgetting the version that RB proved with Loday. The end result there is a cat^n group and hence a model of the n+1 type of the 'union'/colimit. This leads to the non-Abelian tensor product and that has been a great idea, but they do not relevant to the unstable homotopy groups of spheres at least as far as I know. $\endgroup$ Commented Sep 24, 2010 at 6:43
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2$\begingroup$ Hmm, good question. The problem I think moves from calculating homotopy groups of spheres to calculating colimits of cat^n groups. I don't think there is a very big literature on cat^n groups (but I note there is a lot on the nonabelian tensor product - RB has collected them on his website) and tools to deal with them. $\endgroup$– David Roberts ♦Commented Sep 24, 2010 at 8:28
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3$\begingroup$ @ Harry I will do but it will be a bit later on as I have to think about how best to answer e.g. to look up one of RB and J-LLs examples! $\endgroup$ Commented Sep 24, 2010 at 8:33
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5$\begingroup$ I suppose that the theorem in question turns the computation of, say, $\pi_{200}(S^2)$ into mere algebra: it tells you how to write down generators and relations for some complicated algebraic object called a cat_n group and then you just have to learn how to extract your answer from that. $\endgroup$ Commented Sep 24, 2010 at 13:41
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2$\begingroup$ @Tom My own take on it might be a bit different. There is a curiosity in knowing the homotopy groups of spheres, but that is partially a curiosity in studying the way in which the algebra reflects homotopy structure, so modelling the homotopy 200-type of $S^2$ is perhaps in the long run more central to the understanding of the total homotopy type of $S^2$, than 'merely' knowing the value of $\pi_{200}(S^2)$. The $cat^n$ group ($cat^{200}$-group of $S^2$ would contain an awesome amount of information on certain homotopy operations. I cannot start to think what they might be! $\endgroup$ Commented Sep 24, 2010 at 18:02
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I think it is important to remember that the Brown-Loday theorem concerns colimits of cat-n groups obtained from an n stage filtration of the underlying space. Moreover, cat-n groups can only provide information on the n-type.
So, if you wanted to compute pi_200 of the two sphere, you would need a cat-200 group (at least). And if you wanted to apply the Brown_Loday VK theorem you would need a 200 stage topological filtration of the two sphere. Any thoughts ? or do I have this wrong?
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2$\begingroup$ @Carl. Hi. No you are exactly right, but pessimistically so. :-) (See my reply to Tom above) The theory has not been developed that far as to be able to see the ways of extracting information from the gadget. There is probably too much data in the model and one has to see how to use algebraic methods to prise out of it information that is needed. That is not that unusual in algebra. With some of the gigantic simple groups there is a lot of energy that goes into, for instance, understanding what subgroups they have and that sort of effort leads to new tools for use elsewhere. $\endgroup$ Commented Sep 24, 2010 at 18:12
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2$\begingroup$ I feel this is a question of horses for courses. If you wanted to compute pi_200 of the 2-sphere this way you would need a 200-cube structure on the 2-sphere, e.g. from 200 subspaces, and it is not so clear how to get this. However SX does have a convenient triad structure, with two cones, and this is used in classical homotopy theory. The traditional SvKT for the fundamental group does not compute the fundamental group of the circle. The HHSvKT computes some things not otherwise computable, e.g. some nonabelian n-ad groups. Better to look at what these do rather than don't do! $\endgroup$ Commented Sep 27, 2010 at 9:42
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2$\begingroup$ Carl writes
cat-n groups can only provide information on the n-type'. I wonder about the wordonly'! The HHSvKT computes (when it works) the n-type of a colimit from the n-types of the pieces and the gluing information. Sometimes (this has been done for cat$^2$-groups) this allows a lot of information on homotopy groups of the colimit, Whitehead products, composition operators, ... Is not that amazing? Also not many have worked on these gadgets. $\endgroup$ Commented Sep 27, 2010 at 20:04 -
1$\begingroup$ This question really goes back to the 1932 ICM at Zurich when E Cech gave a seminar on higher homotopy groups.,. Their abelian nature suggested that they were not a suitable generalisation to higher dimensions of the fundamental group. We now know that seems to require higher homotopy groupoids of structured spaces. As homotopy groups gradually became an important subject, the other idea became forgotten! $\endgroup$ Commented Jun 8, 2020 at 17:19