What makes an infinite loop space an interesting object of study for homotopy theorists? The reason I ask this question is that I found a lot of results treating the question of whether a given space is an infinite loop space. So it seems that the property of a space being homotopy equivalent to an infinite loop space opens totally new possibilities and techniques to study the space. I would glad if anybody could take some time to give an overview about the most well-known directions one can study a space recognized as infinite loop space.



Why study infinite loop spaces?
I just wrote [an answer to the other thread](https://mathoverflow.net/a/368416/11540), and can expand it into an answer here, about infinite loop spaces instead of just loop spaces.


As mentioned there, spaces of the form $\Omega^\infty \Sigma^\infty X$ contain a great deal of information that helps when computing the stable homotopy of $X$, and, of course, spaces of the form $\Omega^\infty X$ are the zero-spaces of $\Omega$-spectra (and, if the latter is a ring spectrum, then the former is a ring space, in particular a ring object in the homotopy category of spaces).


As described in Adams' book [Infinite Loop Spaces](https://princetonup.degruyter.com/view/title/521833?tab_body=toc), for spaces of the form $\Omega^\infty X$, we have many more tools at hand for computing homotopy and homology, e.g., the infinite loop structure provides invariants based on homology operations including Araki-Kudo and Dyer-Lashof operations. On page 24, Adams describes the use of infinite loop spaces in the proof of the Kahn-Priddy theorem.


Furthermore, Adams describes how the study of infinite loop spaces encompasses the study of generalized cohomology theory (including K-theory and cobordism), and has applications (via classifying spaces like $BTop$) to the geometry of manifolds, units in cohomology rings, and algebraic K-theory.


May's [Geometry of Iterated Loop Spaces](https://www.math.uchicago.edu/%7Emay/BOOKS/gils.pdf) built on this, first by laying out the precise algebraic structure of $n$-fold loop spaces (including infinite loop spaces), then proving the recognition principle, and finally (in chapter 15) deriving practical consequences including spectral sequences, Bott periodicity, and homology operations.


So, to summarize, once you know that a space is an infinite loop space, you have tons and tons of tools at your disposal for carrying out the kinds of calculations homotopy theorists love.


