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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.

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    A silly comment perhaps, but we really care about when spaces are loop spaces and this gives us some machinery to do this. Along with related theorems, such as the approximation theorem, this actually help us to calculate the homology of some things (e.g. loop spaces) that may otherwise be difficult. There is probably a lot more said in 'The Geometry of Iterated Loop Spaces' up on Peter May's page2012-06-28
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    @JuanS: Thanks for your input. Your examples are helpful; if you have enough time, could you perhaps make a (slightly more detailed) answer? I've read (parts of) The Geometry of Iterated Loop Spaces, but unfortunately, as soon as the author starts to speak about the intent of the recognition principle (section 15), he also uses more advanced notions such as spectras, which are only mentioned twice before and not much used (and that I don't really know).2012-06-29
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    I think that was a big motivation though, as infinite loop spaces are a special kind of spectrum. One can also study homology operations on $k$-fold loop spaces as in May, et al, The homology of iterated loop spaces. It also brought to the forefront the importance of $E_k$ algebras which have many applications, mostly advanced, though.2012-06-29
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    OK, thanks. I'll study spectra then :)2012-06-30
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    @nik: Hopefully an expert will add something more! Here is the wiki link for spectra: http://en.wikipedia.org/wiki/Spectrum_%28homotopy_theory%29. Note that the definition of spectrum given there is sometimes called a prespectrum, with a spectrum having the requirement that $E_n \to \Omega E_{n+1}$ is homotopy equivalence. There is a functor from prespectra to spectra that involves infinite loop spaces2012-07-01
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    ... although keep in mind that there are perhaps half a dozen different models for spectra; one person's spectrum is another person's prespectrum. The point is that they all yield the same homotopy category. They all have their pros and cons, of course. I'd recommend Adams' "blue book" pt. 3 ("Generalized Homology and Homotopy" or something like that) for a nice introduction.2012-07-02
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    @AaronMazel-Gee: who calls those prespectra?2012-07-12
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    @SeanTilson: In LMS and EKMM, objects for which the adjoint structure maps are not homeomorphisms are called prespectra.2012-07-15
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    @TylerLawson: His comment seemed to imply that there were people who called the objects with adjoint structure maps homeomorphisms prespectra. This is what I was curious about. Perhaps that was not an intended interpretation.2012-07-21

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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.