We’d like to discuss the homotopy groups of spectra, i.e. what they are, and some very basic things about them.  In the process we will make some vague categorical statements. Hopefully, along the way it will be clear why the category of spectra is in some sense the natural place to do stable homotopy theory.

First of all, we’re going to make some assumptions:

1. All spaces are pointed CW complexes, all maps take base point to point, and all homotopies are homotopies of pointed maps (that is, given a homotopy $H:X\times I\to Y$ such that $H(0,x)=f(x)$ and $H(1,x)=g(x)$, we have that $H(t,x):X\to Y$ is a pointed map.

2. As clearly follows from above, we’ll be working with pointed CW spectra.  We could just as easily work with spectra of simplicial sets.

Now towards defining homotopy groups of spectra, we note that there are always group homomorphisms $\pi_{n+r}(E_n)\to\pi_{n+r+1}(E_{n+1})$.  How can you be so sure, you say?  This is how we can be so sure:  Say we have some map $f:S^{n+r}\to E_n$ which defines a homotopy class.  Then we can define $\Sigma f:\Sigma S^{n+r}\cong S^{n+r+1}\to\Sigma E_n$, where in general, since $\Sigma X=X\wedge S^1$, we can just takse $\Sigma f=f\wedge 1$ (we assume knowledge of the smash product here, especially its construction, but more information can be found at http://en.wikipedia.org/wiki/Smash_product).  This last map $\Sigma f$ clearly defines an element of $\pi_{n+r+1}(\Sigma E_n)$ so by composition with the structure map $\sigma_{n+r}:\Sigma E_{n+r}\to E_{n+r+1}$ we obtain a representative of some class in $\pi_{n+r+1}(E_{n+1})$. Thus, for each fixed $r$ we have a diagram in the category of groups indexed by $\mathbb{N}$, the natural numbers (corresponding to $n$, $n+1,\ldots$).  Since we are in the category of groups, a cocomplete category, we know that the colimit of this diagram exists.  So for fixed $r$ and spectrum $E$ we define $\pi_{r}(E)$ to be $\mathrm{colim}_n\pi_{n+r}(E_n)$.  Note, this colimit exists even for negative $r$ even if we ignore lower degrees where the groups may not exist.

Now, what do we mean when we ask about the stable homotopy groups of a space $X$?  Well, we mean precisely what is discussed above, that is, we mean the elements of homotopy group that remain after an arbitrary number of suspensions.  For instance, the $r$th stable homotopy group of spheres is defined to be $colim_n\pi_{n+r}(S^n)$ but that is precisely the $r$th homotopy group of the sphere spectrum $S$ where $S_n=\Sigma^nS_0=S^n$.

We wish also to have relative homotopy groups of spectra.  This follows directly from our knowledge of relative homotopy groups of spaces.  As we’ve seen in earlier posts, we can just define subspectra as spectra which are level-wise contained in another spectra with natural structure (suspension) maps.

We go ahead and define relative homotopy as

$\pi_{r}(E,A)=\mathrm{colim}_n\pi_{n+r}(E_n,A_n)$

where $A$ is some subspectrum of $E$. The details work out similarly to the above.  Note that for each fixed $r$ and  we have a long exact sequence in homotopy, the one for homotopy groups of spaces.   The diagram we are taking the colimit of is called “filtered,” and filtered colimits are always exact (this is not hard to prove), so we can take the colimit of the LES along $n$ again, which yields the desired LES in the homotopy of spectra.