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.
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11$\begingroup$ It's too late here to write a proper answer, but I strongly recommend reading (or at least skimming) Adams' book titled appropriately Infinite loop spaces. It's rather old but it still contains a lot of great material and it is a good introduction to this circle of ideas. $\endgroup$– Denis NardinCommented Jul 17, 2020 at 23:20
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4$\begingroup$ @DenisNardin's reference: Adams - Infinite loop spaces. As no expert on the subject, I can't say for sure, but this question seems far too broad to be given a reasonable answer in the MO/SE format. $\endgroup$– LSpiceCommented Jul 18, 2020 at 1:13
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2$\begingroup$ One way is to say that why they are important is we can define negative homotopy groups of such spaces. As we know homotopy groups of CW complexes X contains a lot of information about X. So we can expect that for such spaces, we have more algebraic invariant attached to it and homotopy theory will be more effective in such case. $\endgroup$– SunnyCommented Jul 18, 2020 at 3:38
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3$\begingroup$ this link may help you a little mathoverflow.net/questions/365447/loop-spaces-motivation $\endgroup$– SunnyCommented Jul 18, 2020 at 3:48
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1$\begingroup$ Not just a motivation, but $A_\infty$ structure is relatively simple to describe, so it was surely a starting point. In your context, it is helpful to think of infinite loop spaces as ‘linear objects which supports stable invariants such as cohomology (in fact, they are higher version of abelian groups), contrary to general spaces being ‘nonlinear’ and supports no algebraic stricture itself. $n$-fold loop spaces are intermediate hierarchy between these linear and nonlinear situations. Another modern short explanation can be found in Lurie’s Spectral Algebraic Geometry, p.50-51 $\endgroup$– nrkmCommented Jul 22, 2020 at 6:51
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2 Answers
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I just wrote an answer to the other thread, 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, 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 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.
answered Aug 6, 2020 at 11:57
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$\begingroup$ what is a zero-space of a $\Omega$-spectrum ? (I suspect the question is embarrassing but google refused to find an answer) $\endgroup$ Commented Aug 16, 2020 at 22:29
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$\begingroup$ So, a spectrum is a sequence of spaces $X = (X_n)$, one for each natural number, plus some extra data relating $\Sigma X_n$ and $X_{n+1}$. By "the zero space" I simply mean the space $X_0$ (here space could mean topological space or simplicial set, depending on what type of spectra one is working with). $\endgroup$ Commented Aug 16, 2020 at 22:33
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There is so much much more. For one historical starting point among many, you see that many spaces of interest are infinite loop spaces and that tells you how to calculate things about them. For just one example, to add to David's parenthetical clause, almost everything we know about characteristic classes for topological bundles comes from the infinite loop structure of BTop. This is very concrete and calculational and tells us geometrically about topological cobordism. At another extreme, knowing that algebraic K-theory is given by E infty ring spectra is the starting point for derived algebraic geometry. I could go on for pages and pages. The emerging equivariant story is even richer and promises far more to come.
answered Aug 6, 2020 at 13:52