How to get `K(A,n)`, for any finitely generated abelian group `A`?

[jcguu95]

It seems that the classifying spaces K(A,n) are only implemented for A = Z or Z/2Z. How do I construct K(A,n) in general (note: we also need to build an effective chain complex efhm for it)?

[Alizter]

If your A is finitely presented, why not just invert all odd primes and then you will have a cartesian product of K(Z,n) and K(Z/2Z,n). I don't know if this is possible in Kenzo, but just an idea. I reckon it would be possible since a cartesian product of spaces is quite a simple operation.

[jcguu95]

You can do cartesian product. But what do you mean by inverting odd primes? I'm a novice in homotopy theory, but I know some math - would you mind elaborating a bit more?

[Alizter]

Actually I noticed what I wrote isn't strictly correct. There can be Z/2^nZ that will be missed. The way Kenzo calculates homotopy groups is I think via the Hurewicz theorem. If you have K(Z,n) and K(Z/2Z,n) that can tell you enough about the mod 2 cohomology and homology for you to piece together some info about the homotopy groups.

In general, K(A,n) is not interesting from the point of view of homotopy groups since we know all of them, but the homology and cohomology is going to be interesting. When consider only mod 2 cohomology, you have more structure than in the Z case allowing you to get some help with computations. The other primes are generally much more sparsely located so the mod 2 case is the main focus.

[Alizter]

I wrote this a bit earlier, but then I realised my mistake. I will share it anyway so you can get a feel for what I was going for:

When we study spaces in homotopy theory we are interested in homotopy invariants like the fundamental group and higher homotopy groups. These are invariant since we have ways of transforming a space, known as homotopy equivalences, that leave these invariants untouched. This is essentially the space version of an isomorphism.

Now the homotopy groups of a space are like an "algebraic shadow" of a space. It tells us a lot of interesting information about that space and how it is built up. In general, the algebraic world is very amenable to computation, which makes it a very desirable outcome. If you can relate some object to an algebraic object, then you have all sorts of tools at your disposal to study the algebraic object and infer something about the original object.

Now homotopy groups are abelian groups, except at the first point, the fundamental group. In that case, typically we ask for 1-connected spaces, so that there is no interesting fundamental group and all the homotopy groups become abelian. In that case we have an invariant that turns spaces into abelian groups.

When working with abelian groups we can imagine inverting certain elements. This generalises the classical construction of the rational numbers Q. You start with the integers Z and you say, "I wish I could invert n". You simply write 1/n and then some rules for how to multiply it and you get Q. The idea is to freely add all the inverses that you don't have.

The interesting part of this process is that if you invert everything but a certain prime p, you will get a group where the elements made up of a prime don't have inverses, in that sense they stick around. In the case of Q, we inverted all the elements so only the infinite order elements stuck around. In the case of Z where anything but p is inverted, only the elements which have order p^n for some n stay, whereas all the other primes get destroyed.

So we have a way of decomposing algebraic objects so that we can study them prime by prime. It turns out there is a way to put them back together. Wouldn't it be nice if we could do this "localisation" business for spaces before we pass to the homotopy groups? It turns out we can, and it is a very powerful technique. We can add certain cells to a space that can destroy any information we pick, allowing us to filter out only the information at a prime p. It is such a good filter that we also have a way of sticking these localised spaces back together to get the original space!

Now, for Eilenberg-MacLane spaces this localisation business is far too complicated. They are a sort of "mirror image" of the abelian group world. You take a group A, and product a space whose nth homotopy group is A. When you localise a space to keep only the p-primary information, you are actually only localising the group A, which we already know how to do!

So my suggestion above is to do just that. If you are interested in the 2-primary information of a space, just localise it with respect to 2. This means you take you abelian group A, use the fundamental theorem of abelian groups to get it to be a direct sum of some copies of Z and some copies of Z/q^nZ for various primes and powers. Then, with the caveat of being a finite A (so no Zs), you only pick the Z/2^nZ. Generally in homotopy theory, 2-primary parts are the hardest to calculate and probably have the most interesting structure.

If you are interested in some infinite abelian groups, then you need to tensor with the rationals Q and consider the rationalisation of a space. For K(A,n) this amounts to tensoring A with Q which will destroy all the finite information. I don't know if Kenzo can do this, but rational homotopy types have other nice computational tools.

Anyway, once you've weeded out all the 2-primary parts of what you are interested in, you will see that we only care about K(Z/2^nZ, m). The invariants we are interested in are no longer homotopy groups since K(A,m) can easily be calculated (hint its just A), what we are interested in is homology and cohomology.

Cohomology reduces down to studying maps K(Z/2^nZ, m) -> K(Z,k) which can have a lot of structure. There is a whole lot of literature about just this case. A good reference is the book by Mosher and Tangora called "Cohomology Operations and Applications" in homotopy theory.

From what I remember, homology but can be calculated as repeated smashes of K(Z,k) and K(Z/2^nZ,m). That smash has a highly connected map to the EM space of the tensor product of those two at level k+m which can be used to calculate homology.

[jcguu95]

Wow! Thanks!

What if I want to calculate the space of homotopy maps from X to Y?
Currently, Kenzo can handle when X is a sphere by taking Whitehead tower (chap 21 of doc) and use Hurewicz theorem.

However, for general X, this procedure would involve a map from X to K(pi_k(X), n) for some k, n. That pi_k(X) could be any abelian group. How would this be done when only the K(Z,n)'s and K(Z/2,n)'s are provided?

[adlai]

Thank you @Alizter for fearlessly derailing this Issue into a mathematics lesson, probably much needed for dropouts like me who are autodidact both in practice and theory.

Now the homotopy groups of a space are like an "algebraic shadow" of a space. It tells us a lot of interesting information about that space and how it is built up. In general, the algebraic world is very amenable to computation, which makes it a very desirable outcome. If you can relate some object to an algebraic object, then you have all sorts of tools at your disposal to study the algebraic object and infer something about the original object.

I get triggered by metaphoric language; how did you choose "algebraic shadow" rather than the possibly-wrong word projection ? I always envisioned the structures resulting from further consideration after learning the fundamental groupoid to be projections of the higher-dimensional spaces into the lower-dimensional territory of finitely presented groups.

If this is the wrong place for such questions, please say so; otherwise, the website makes it surprisingly awkward to ignore me.

[Alizter]

I get triggered by metaphoric language; how did you choose "algebraic shadow" rather than the possibly-wrong word projection ? I always envisioned the structures resulting from further consideration after learning the fundamental groupoid to be projections of the higher-dimensional spaces into the lower-dimensional territory of finitely presented groups.

I chose the word "shadow" with the geometric picture in mind. It is the essence of something that in some way loses information. So you could think of it as a projection onto a lower dimensional space.

If you imagine a cube in a room with a light hanging from the ceiling, it will product a shadow. You can rotate the cube in various different ways and observe that the shadow shifts shape between various polygons. Using only the shadow you may be able to derive some properties of the cube, such as the fact that it is connected, how many edges etc. The issue is that you will never have the complete picture of the original space. For example, what colour was this cube? The idea with an algebraic invariant is the same, we have something very complicated (in this case a space) and we want to turn it into something computable like an algebraic object.

Algebraic invariants are only really good when they are different. If two spaces have the same homotopy groups, you need to know something extra in order to conclude that they are really different. This extra fact is in essence a way of gluing these groups together to make a space. In algebraic topology we have the Whitehead theorem. If I have a map f : X -> Y and since homotopy groups are functorial, we have group homomorphisms pi_n(f) : pi_n(X) -> pi_n(Y). Whitehead then tells us that f is a homotopy equivalence when all of those group homomorphisms are isomorphisms.

You mention the fundamental groupoid, this is in some sense an enhancement of the fundamental group. The simple picture is if I take a space, add all the points, and then add all the paths between those points I get the fundamental groupoid. The difference with the space itself is that a space might have multiple homotopies between a given path, whereas the fundamental groupoid only has one. It is in a sense a truncated space. The fundamental group says what if we chose a single basepoint, and considered all paths to and fro. This has advantages since groups have a lot of theory, and when a space is path-connected it is equiavalent since you can arbitrarily change basepoint. However in disconnected spaces there are problems. In those cases, people generally study the connected componenents separately.

I also don't know if this is an appropriate place to share these discussions, but until somebody says otherwise I don't think there is any harm.

[Alizter]

Wow! Thanks!

What if I want to calculate the space of homotopy maps from X to Y? Currently, Kenzo can handle when X is a sphere by taking Whitehead tower (chap 21 of doc) and use Hurewicz theorem.

The answer is "it depends". Homotopy groups are a special case of "what are all the distinct points of the mapping space S^n -> X". So generally mapping spaces are quite wild objects. Kenzo is doing the method of "killing lower homotopy groups using the Whitehead tower", which is plausible but very quickly blows up in complexity.

However, for general X, this procedure would involve a map from X to K(pi_k(X), n) for some k, n. That pi_k(X) could be any abelian group. How would this be done when only the K(Z,n)'s and K(Z/2,n)'s are provided?

Generally the spaces we care about are CW-complexes which is maths-speak for glued from n-spheres in a special Lego-like way. This means that homotopy groups of such spaces can be calculated from the homotopy groups of the pieces, modulo some important finer details. Therefore the homotopy groups of spheres are the only case that is considered in detail.

For homotopy groups of spheres, we have things like Serre's finiteness theroem which says that most homotopy groups of spheres are in fact finite. In that case, the procedure I mentioned earlier becomes easy and you can focus just on the 2-primary information. Mostly the 2-primary information is what is calculated in practice. As I said before, you do all the p-primary calculations separately in these sophisticated computations.

[jcguu95]

How do you compute $\pi_7(S^3)$ then?

Currently, we can compute $\pi_6(S^3) \simeq Z/12Z$ exactly because the lower homotopy groups only involve $Z$ and $Z/2$. Take a look at the code:

(setf s3 (sphere 3))
;(homology s3 3)
(setf fcc3 (chml-clss s3 3))
(setf fbr3 (z-whitehead s3 fcc3))
(setf x4 (fibration-total fbr3))
;(homology x4 4) ;This computes \pi_{4}(S^{3})
(setf fcc4 (chml-clss x4 4))
(setf fbr4 (z2-whitehead x4 fcc4))
(setf x5 (fibration-total fbr4))
;(homology x5 5) ;This computes \pi_{5}(S^{3})
(setf fcc5 (chml-clss x5 5))
(setf fbr5 (z2-whitehead x5 fcc5))
(setf x6 (fibration-total fbr5))
(time (homology x6 6)) ;This computes \pi_{6}(S^{3})

Notice that the key functions are #'z-whitehead and #'z2-whitehead. Those functions are used to build the next stage in the whitehead tower. One should use #'z-whitehead if the current homotopy group to be killed is $Z$, while use #'z2-whitehead while it's $Z/2$.

To get to the 7th homotopy group, we need something like #'z12-whitehead, since $\pi_6(S^3)$ is $Z/12$. However, #'z12-whitehead isn't implemented yet in kenzo. In fact, I think we need $K(Z/12Z, n)$ to implement that.

And to push this higher, we need $K(A,n)$ for all possible homotopy groups of spheres $A$ (see the table).

[Alizter]

So using what I said above, we know that Z/12Z = Z/4Z + Z/3Z so we just need to kill z4 and z3. I'm unsure if z2 and z4 can be generalized together. You can also just ignore the 3-primary part since it won't affect the 2-primary part. Although I'm unceratin if Hurewicz still works locally.