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In algebraic topology one thing you might do is compile a table of known homotopy groups of spheres. These are great for flashing at people in order to get them interested in the subject but they contain very little information really. What they ought to do is link with the literature, so that people can check them etc.
Now that we live in the future, we don’t have to compile these on paper, we can have a digital place to store them. nlab seemed like a good candidate at first, however it has some drawbacks. The main one being there really is no “spreadsheet” like behaviour for the nlab.
Over the summer I had compiled a table(s), past wikipedia’s range, however I quickly realised that it would be difficult to go past the 20-stem. This is because most literature focuses on the 2-primary part, the hardest part, and almost no literature discusses putting the pieces together. There are a few books detailing ’random’ facts about which p’s are important but I have never seen a single place these have been written down. It is not easy to quickly find results without being an expert.
Now my table isn’t perfect, it doesn’t include a lot of important detail like generators and j-homomorphism. However it could be a good start for compiling a really linky table on the nlab. If someone can suggest a way to display these spreadsheets on the nlab, I would be grateful.
Another problem: A lot of the literature is old(er than me atleast). Toda’s book “Composition methods … something something spheres” details nicely all the techniques he used to crack out these results, however it is like reading French for me. I have to slowly translate it into the more modern language I am familiar with. Again, nlab would do this and maintain this better.
Also if anybody wants to improve these tables drop me an email, it’s my username at gmail.com
I think the nLab would be a great place to collect together information about the calculations about homotopy groups of spheres. There could be a page per group/collections of groups, etc, with some summary page. Many possibilities no doubt. I’ve long wished to work through parts of Toda’s book. A bit like the initiality project, this is something which is very useful and interesting for many people, but is tedious/tricky to write down oneself, so seems the perfect thing to do as a community.
There are also numerous perspectives. For instance, the higher categorical. I have no idea why from the structure of a 6-groupoid: one can something about and say something about, but it’s a mystery (to me at least) what is going on beyond that.
Also the finite. I’ve not seen anybody try to calculate homotopy groups of spheres (beyond the first couple) using finite models, but I suspect it is very possible to do much of algebraic topology in this setting. E.g. I suspect one would have a version of the Serre spectral sequence.
@Richard the Kenzo program uses “finite” models. It calculates homotopy groups by killing lower homotopy groups with the whitehead tower and crunching the simplicial complex that arises. This is a very exponentially algorithm however and hasn’t been able to do anything new. One of the hopes with HoTT is that the types are automatically datatypes which means in the (far) future one could sit down and algorithmically compute the groups, a bit like Mahowald does but this at the moment is complete nonsense until HoTT sorts out issues with computability of univalence etc.
Interesting! By finite, I meant things like the pseudocircle; does Kenzo use finite topological spaces in this sense?
this at the moment is complete nonsense
Hehe, I think it’s a great quest mathematically (i.e. I think a lot of great stuff will emerge as a by-product), but I would indeed be astonished if HoTT can help with computation beyond what a tailor-made tool could achieve :-).
Oh, it uses simplicial complexes (in a really convoluted way imo) written in lisp. I have never seen anyone do anything with finite topological spaces in order to compute homotopy. I would argue that its the wrong kind of information.
I think one of the big progressions that will happen is that haskell (or a haskell like language) will get dependent types ~2020, which on itself will be a massive improvement. Then people will start modifying it with universes etc. All along the way optimisation will be found. I reckon in around 10 years time we will be able to “formalise” HoTT book material in a haskell like language.
Why ‘the wrong kind of information’? It gives a finite and complete description of the simplicial complex and of its homotopy type, by encoding the poset of the cells. Very sensible and intuitive.
I have never seen anyone do anything with finite topological spaces in order to compute homotopy.
Indeed, I agree that there is little literature on this, but it is something I have been pondering for a number of years, just not something I’ve ever had time to properly investigate. Finite topological spaces are remarkable, you would be surprised what one can do with them :-). I really suspect that everything in traditional algebraic topology has an analogue for finite topological spaces, just nobody has ever systematically explored this.
I would argue that its the wrong kind of information.
I agree with Tim here, it is a different kind of presentation of the homotopy type, but it is intuitive in its own way. The subject is more combinatorial in flavour, yet also not so far from category theory.
I reckon in around 10 years time we will be able to “formalise” HoTT book material in a haskell like language.
I don’t know about how soon it will come (if you’d asked me 10 years ago, I probably would have guessed that Haskell would have dependent types by now), but I agree that it seems likely that Haskell will get them eventually, and that this should indeed lead to it being much more practical to use them, as well as providing some speed up. But still I wouldn’t have thought one could ever match more lower level computation; there is an enormous amount of overhead with dependent types, at least in any current implementation.
Have a look at: Jonathan Barmak, _ Algebraic Topology of Finite Topological Spaces and Applications_, Lecture Notes in Mathematics Vol. 2032. Springer (2011).
Thanks Tim! That book is great, but what I have in mind goes a bit further: I think there should be Serre/Adams/May spectral sequences, I think there should be a chromatic picture, etc. I.e. if one sort of follows the post-Serre evolution of algebraic topology, I think there should be an analogue of more or less everything in the finite world. And this could be really interesting.
@Tim, that’s an interesting book, I will have to give it a good skim later. I am surprised I had not came across it before.
@Richard How would you construct Eilenberg-MacLane spaces?
Thanks for the question! I’ll have a think about it!
(I should say that by finite I really meant Alexandroff, i.e. where the theory is equivalent to that of preorders.)
Jonathan has continued working on many of the ideas there since he wrote those Lecture Notes. The book was based on his thesis.
There is a group doing interesting stuff at UBA led by Gabriel Minian.
The paper on minimal finite spaces by Barmak and Minian:
may be of interest.
(Thought: as often happens this is going off thread! I will continue it if needs be in another even more relevant thread.)
Maybe I’m not following this thread too well, but regarding #10: isn’t it enough that nerves of posets model all weak homotopy types? Given that Richard said “by finite I mean Alexandroff”, and we just apply the McCord theory.
Re #15; Yes, I think that is correct. But I also think one should be able to do it ’purely within the Alexandroff world’. I’ll try to elaborate when I get the chance.
I am going to ressurect this thread so we can get started. I propose an article called “table of homotopy groups of spheres” which will contain a table with links to seperate articles as entries. We can start with and give the classical argument for it. (How would we name such an article).
I think it would make sense to table the groups like . We should be able to write out most “classical” calculations that can be found in algebraic topology textbooks (Hatcher, Bott and Tu, etc.). I think Ravenel would have some things to say on the first few unstable groups too. The next big reference to disect would be Toda’s book, of which I have a copy. The main problems here are the outdated notation and “fill in the details style”. Since this book has been written I am sure many people have improved upon results so it is a matter of finding them.
I think Toda pretty much pushes up against the limit of what can be done by hand, I am fairly certain the ’s past 30 are done with computer assistance (Mahowald?). But I believe the techniques are very similar to the ones for lower .
I don’t think we have to mention spectral sequences at all for groups up to . But I think it would be good to improve what we have on the nlab already. I think Toda is using the EHP ss, but as a long exact sequence due to low . I could be wrong.
Does this sound like a sound plan?
You don’t need to propose such actions, you should feel invited to go ahead. It is evidently worthwhile and classical, completely uncontroversial material.
The canonical title for the entry would evidently be “stable homotopy groups of spheres” (where, it seems, you will want to create a section “Examples”). Currently this is redirecting to homotopy groups of spheres but if you get serious about writing on the stable case you just remove the redirect and create the new entry.
Generrally, there is no need to make big a priori plans for entries, nor do these tend to work out as expected anyway. The way to proceed is by convenient increments:
Whenever you have anything to add, you look for the closest existing entry where it could be included, or if that does not exist, or does exists but is already too crowded, then you create/split off a new one and go ahead.
Hi Ali, I would be happy to help out with this as much as time allows, I have long wished to work through Toda’s calculations. I unfortunately do not have access to a copy of Toda’s book, but I suppose we can get around that one way or another. I can also have a go at contributing computer code, if I know what to compute!
Bit of a tricky question what to name the pages for the individual homotopy groups (I definitely agree that this is a good structure). In this case I think it is best to avoid mathematics, so that for example the title looks OK in the browser. Maybe ’third homotopy group of the 2-sphere’, or whatever: long-winded, but at least accurate. Then we have redirects for shorter forms, maybe say pi_3(S^2) verbatim, maybe using a unicode pi symbol, etc.
@Urs I want to focus on the unstable case more as I think the stable case is pretty well documented on the nlab. The homotopy groups of spheres article really only talks about the stable case.
@Richard I have a copy of Toda’s book and access to a photocopier so drop me an email any time.
Ali, all the better, then just create a subsection “Examples” in the existing entry and get going.
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