Last time, we defined two functors and on the category of pointed topological spaces and (base-point preserving) homotopy classes of base-point preserving continuous maps. We showed that they were adjoint, i.e. that there was a natural isomorphism
We also showed that is naturally an H group, i.e. a group object in , for any . So, given that we have a group operation , it follows that is naturally a group for each .
I should perhaps briefly explain that the category admits products. The product of is just . It also admits coproducts: the coproduct of is the wedge sum of with the points identified to one (base)point. (This really should have been mentioned at the beginning. Otherwise, the notion of a group or cogroup object doesn’t make sense.)
It follows that the functor
can be viewed as taking values in the category of groups. Thus, must be, by Yoneda nonsense, a cogroup object in , or what we called an H cogroup. It can be checked that the cogroup structure on , up to homotopy, of the form
that sends the path in to the path in the first copy of (in ) followed by in the second copy.
0.1. Homotopy groups
Let be the -sphere with some basepoint. Then I claim that is just (with some basepoint).
The easiest way for me to see this is to think of (with the basepoint being ) instead of . Indeed, we have a homeomorphism
When we suspend , we take the product , and then collapse the two ends to points. We then collapse the entire product and collapse that to a point. The total equivalence relation that we end up quotienting out by is just that which identifies all of to a point.
So it is clear that the suspension of is . This implies the claim about . In particular:
Proposition 1 is an H cogroup.
So the sets of homotopy classes of continuous maps are actually groups, because is an H cogroup, in turn because is a suspension.
1. is abelian
There is a very good reason for the higher homotopy groups to be abelian. A visual illustration of this, which I recommend looking at, is given in Hatcher’s Algebraic Topology. I want to discuss the categorical picture.
Fix . We know that suspension and loop space are adjoint functors on the category in which we are currently interested. In particular, we have that
by the adjoint property.
The point is that this has two natural group structures. On the one hand, is an H cogroup. On the other hand, is an H group by catenation of paths.
We now appeal to a general principle to show that is abelian.
Proposition 2 Let be an H group and an H cogroup. Then the two group structures on
are equal, and is an abelian group under this structure.
Proof: The idea is that we have two “independent” ways of defining the group law on . Namely, one has the “unit” for the two group laws: it is just the constant map whose image is the basepoint of . Moreover, the independence statement states that the two operations distribute over each other. This is intuitively clear because one operation depends on the action of and one operation depends on the action of . But it should be formally checked.
Now, the Eckmann-Hilton argument implies that the two group operations are the same and abelian.
2. The Eckmann-Hilton argument
We will prove this more general fact from the following even more general argument:
Proposition 3 (Eckmann-Hilton argument) Suppose is a set and are two operations which have a common identity element and which are mutually distributive, i.e.
Then are equal, and the operation is commutative.
The proof of the Eckmann-Hilton argument is essentially formal manipulation. We write:
Hence, from that formal manipulation, are equal. Now let us prove commutativity:
by what has already been proved.