Last time, we were discussing the category ${\mathbf{PT}}$ whose objects are pointed topological spaces and whose morphisms are pointed homotopy classes of basepoint-preserving maps. It turns out that the homotopy groups are functors from this category ${\mathbf{PT}}$ to the category of groups.

The homotopy group functors ${\pi_n}$ are, however, representable. They are representable by ${(S^n, s_0)}$, where ${s_0}$ is a base-point; this is equivalently ${I^n/\partial I^n, \partial I^n}$ for ${I^n}$ the ${n}$-cube and ${\partial I^n}$ the boundary. The fact that these are functors to the category of groups is equivalent to saying that ${(S^n, s_0)}$ is a cogroup object in ${\mathbf{PT}}$.

But why should ${(S^n, s_0)}$ be a cogroup object? To answer this question, let us consider a pair of adjoint functors on ${\mathbf{PT}}$.

1. The loop space

Let ${(X,x_0) \in \mathbf{PT}}$. The loop space ${\Omega X}$ (which should really be denoted ${\Omega(X, x_0)}$) is the set of all maps

$\displaystyle (I, \partial I) \rightarrow (X, x_0).$

In other words, the set of all loops at ${x_0}$, or all maps ${I \rightarrow X}$ which start at end at the basepoint ${x_0}$. With the compact-open topology, this is a nice topological space.

This is starting to sound like the fundamental group ${\pi_1(X, x_0)}$, but that’s actually the set of path components in ${\Omega X}$.

${\Omega X}$ is a pointed space. Indeed, the constant path at ${x_0}$ is the basepoint. Moreover, ${\Omega}$ is functorial. Given a map ${X \rightarrow Y}$ of pointed topological spaces, we get a continuous map

$\displaystyle \Omega X \rightarrow \Omega Y$

by composing paths into ${X}$ with ${X \rightarrow Y}$. Since the constant path at ${x_0}$ is sent into the constant path at the basepoint of ${Y}$, ${\Omega X \rightarrow \Omega Y}$ is a morphism of pointed spaces.

But we still need to make ${\Omega}$ into a functor on the homotopy category ${\mathbf{PT}}$. This results from the following easy observation. Given a basepoint-preserving homotopy of maps ${X \rightarrow Y}$, we get a homotopy of maps ${\Omega X \rightarrow \Omega Y}$. So ${\Omega}$ is a functor on the homotopy category.

To summarize:

Proposition 1 ${\Omega}$ is a functor ${\mathbf{PT} \rightarrow \mathbf{PT}}$.

Nonetheless, ${\Omega X}$ is more than just a pointed space. It is also, in a very natural manner, a group object in ${\mathbf{PT}}$. In particular, this means that there is a “group operation”

$\displaystyle \Omega X \times \Omega X \rightarrow \Omega X$

and a “unit”

$\displaystyle \ast \rightarrow \Omega X$

and an “inverse”

$\displaystyle \Omega X \rightarrow \Omega X$

such that the associativity, inverse, unital axioms hold up to homotopy. This is the dual of the idea of an H cogroup discussed earlier.

The group structure is defined as follows. Given loops ${s: I \rightarrow X, t: I \rightarrow X}$ at the basepoint, their “multiplication” is defined by concatenation ${s \ast t}$: first go through ${s}$ (at twice the speed), then go through ${t}$. The inverse sends ${s}$ to its reversal. The unit element is the constant loop (i.e., the basepoint). It is easy to see that there are homotopies

$\displaystyle s \ast (t \ast u) \simeq (s \ast t ) \ast u$

proving H associativity, and so on. These calculations, which are analogous to those for the fundamental group, show that ${\Omega X}$ is a group object in ${\mathbf{PT}}$.

2. The suspension

Now we describe the adjoint. Suppose ${X, Y, Z}$ are topological spaces, where ${Y}$ is locally compact and Hausdorff. Then the obvious map of sets

$\displaystyle \mathrm{Fun}(X \times Y, Z) \rightarrow \mathrm{Fun}(X, \mathrm{Fun}(Y,Z))$

is a homeomorphism if ${\mathrm{Fun}(A, B)}$ is treated as a topological space for any spaces ${A,B}$ by giving it the compact-open topology.

If we treat the outside “${\mathrm{Fun}}$‘s” as simply sets, one consequence of this is that the two functors ${- \rightarrow - \times Y}$ and ${- \rightarrow \mathrm{Fun}(Y, -)}$ are adjoint functors on the category of topological spaces if ${Y}$ is locally compact and Hausdorff.

Now ${\Omega}$ is a kind of ${\mathrm{Fun}}$ but with some restrictions. So the previous paragraph suggests that it might have an adjoint, which would look like a product. But ${\Omega}$ is a restricted form of ${\mathrm{Fun}}$: it is a subfunctor. Thus the adjoint should be a quotient space, perhaps. So it seems.

We now describe this functor. Start by recalling a notion of standard topology.

Definition 2 Let ${X}$ be a topological space. Then the suspension of ${X}$ is the quotient of the product of ${X \times I}$ with ${X \times \left\{0\right\}}$ and ${X \times \left\{1\right\}}$ each identified to a point. The suspension is denoted ${\Sigma' X}$.

So intuitively, one should think of the suspension as pinching off the top edge and the bottom edge of the cylinder ${X \times I}$ to a point (well, two different points).

Since we are in the category of pointed spaces, we need a pointed version of this. The suspension by itself does not have a nice pointed structure to it.

Definition 3 Let ${(X, x_0)}$ be a pointed topological space. Then the reduced suspension ${\Sigma X}$ is the quotient of the ordinary suspension ${\Sigma' X}$ with ${x_0 \times I}$ identified to a point. The basepoint of ${\Sigma X}$ is the equivalence class ${\left\{x_0 \times I\right\}}$.

I claim now that ${\Sigma}$ can be made into a functor on ${\mathbf{PT}}$. First, suppose ${f: (X, x_0) \rightarrow (Y, y_0)}$ is a continuous map of topological spaces. We get a map

$\displaystyle f \times 1: (X \times I, x_0 \times I) \rightarrow (Y \times I, y_0 \times I).$

This map sends the top edge into itself and the bottom into itself. It is clear now that ${f}$ factors through the equivalence relation defining the reduced suspension and becomes a map

$\displaystyle \Sigma f: \Sigma X \rightarrow \Sigma Y.$

Similarly, given a homotopy of maps ${X \rightarrow Y}$, we get a homotopy of maps ${X \times I \rightarrow Y \times I}$ that keep the second coordinate fixed. It is thus clear that we get a homotopy of maps ${\Sigma X \rightarrow \Sigma Y}$. These remarks have proved:

Proposition 4 ${\Sigma}$ becomes a functor ${\mathbf{PT} \rightarrow \mathbf{PT}}$.

The main result we have been waiting for today is that:

Proposition 5 ${\Omega}$ and ${\Sigma}$ are adjoint functors on ${\mathbf{PT}}$. In particular, we have a natural isomorphism$\displaystyle \hom_{\mathbf{PT}}( \Sigma X, Y) \simeq \hom_{\mathbf{PT}}(X, \Omega Y).$

In a sense, this is almost tautological once one thinks through it carefully. Suppose given ${\Sigma X \rightarrow Y}$. Then for each ${x \in X}$, as ${t}$ ranges through ${I}$, we have the path ${t \rightarrow (x, t)}$ traced out in ${\Sigma X}$. So each ${x \in X}$ gets sent to a path ${l_x}$ in ${Y}$. This goes to a path in ${Y}$. Since ${\Sigma X \rightarrow Y}$ is a base-point preserving map, we see that ${l_x}$ is a loop at the basepoint ${y_0 \in Y}$. Moreover, the basepoint of ${X}$, that is ${x_0}$, is sent to the constant loop at ${y_0}$, because ${x_0 \times I}$ is identified to one point in ${\Sigma X}$. In total, we get a base-point preserving map

$\displaystyle X \rightarrow \Omega Y.$

One works in the same way to get a map ${\Sigma X \rightarrow Y}$ out of ${X \rightarrow \Omega Y}$. Then, one can check that basepoint-preserving homotopies of continuous functions ${\Sigma X \rightarrow Y}$ induce basepoint-preserving homotopies of continuous ${X \rightarrow \Omega Y}$, and vice versa. This is how the proof works.

Next time, I’ll discuss the cogroup structure on the suspension, and finally explain how this connects to the homotopy groups. I’ll also discuss the Eckmann-Hilton argument for showing that the higher homotopy groups are abelian, which comes right from here.