Why is the recognition principle important?

The recognition principle basically states that (under some conditions) a topological space $X$ has the weak homotopy type of some $\Omega^k Y$ iff it is an $E_k$-algebra (ie. an algebra over the operad of the little $k$-cubes). This principle is often quoted as one important application of operad theory (see this MO post for example).

My question, then, is: why is the recognition principle important? What are some typical applications, consequences? I know it has some links to commutativity of loops and things like that, but I'm not quite sure what that entails.

Solution 1:

Calculating homotopy groups in general is pretty hard, at least compared to computing homology or cohomology. In between these two extremes, you can try to compute the stable homotopy of a space $X$. By definition, the $i^{th}$ stable homotopy group of $X$ is equal to $\pi_{i+k} \Sigma^{k} X$ for sufficiently large $k$. This group is denoted $\pi^s_i X$. In fact, the sequence of functors

$\pi^s_i(-) \colon \mathrm{Spaces} \to \mathrm{Abelian Groups}$

forms a generalized homology theory. So these stable homotopy groups can be accessed with many of the same tools and methods as used on ordinary homology.

Now, observe that $\pi_{i+k} \Sigma^k X = \pi_i \Omega^k \Sigma^k X$, so instead of studying $\pi_i^s$, you could study spaces of the form $\Omega^k \Sigma^k X$ for large $k$. As mentioned in the comments, if you take the colimit of these spaces as $k\to\infty$, you get a spectrum, which is the same thing as a generalized homology or cohomology theory.

All of the above is just to say that showing certain spaces are spectra is an example of an important application. Below, I argue that the recognition principle helps you construct such examples.

One simple conclusion from May's delooping theorem is that a topological space with an abelian group structure gives a cohomology theory. In addition, it is often easier to show you have an $E_n$ structure on a space $Y$ for each $n$ than to find a space $X$ such that $\Omega^\infty\Sigma^\infty X = Y$. Usually $Y$ is the classifying space of some symmetric monoidal category. There are other examples, although none come to mind at the moment, of categories with extra structure whose classifying spaces give rise to infinite loop spaces.