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Is there a classification of one periodic cohomology theories? In other words, spaces $X$ such that $\Omega X \simeq X$? For example, $X=*$ and $X= K(G,0) \times K(G,1) \times \dots$ are examples. More generally, if $\{Y_n\}$ is an Omega spectrum $X=\dots \times Y_1 \times Y_0 \times \Omega Y_0 \times \dots$ is an example.

I ask because I have a space (related to stable block bundles) which does not seem like it should be contractible but appears to be its own loop space.

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    $\begingroup$ Is there any additional structure you're putting on these? I'm skeptical that there's a classification of such things without any additional conditions imposed. For what it's worth, a sort of "universal" example can be obtained as the Thom spectrum of the inclusion Z -> Pic(S^0) = Z x BGL_1(S^0). The connective cover of this spectrum is ΩS^2_+, and so it is an E_1-ring spectrum, but I don't think it has more structure than that. (That's one reason why 2-periodic things are nicer: the Thom spectrum of Z = Ω^2 CP^oo -> Ω^2 BU = BU x Z is an E_2-ring spectrum, whose connective cover is ΩS^3_+.) $\endgroup$ – skd Aug 10 at 2:24
  • $\begingroup$ @skd I wasn't aware of any nontrivial examples, so I mostly wanted to know if anyone was automatically boring. Interestingly, it did come up in conjunction with $Pic(S^0)$ which I don't know anything about. Do you have any reference for basic facts about $Pic(S^0)$. Particularly its relation to the J-homomorphism. $\endgroup$ – Connor Malin Aug 10 at 2:43
  • $\begingroup$ You can think of the J-homomorphism as the geometric realization of the (symmetric monoidal) functor N(Vect_R) -> Pic(S^0), where Vect_R is the category of finite-dimensional real vector spaces and linear isomorphisms, sending V to its one-point compactification S^V. There is a weak equivalence N(Vect_R) = Z x BO. Note that Pic(S^0) = Z x BGL_1(S^0), and the J-homomorphism sends the Z to Z (because the one-point compactification of R is S^1, the generator of the Picard group of Sp), and BO -> BGL_1(S^0) sends a bundle to the associated spherical fibration. Not sure of a reference, though. $\endgroup$ – skd Aug 10 at 3:51
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    $\begingroup$ These "1-periodic spectra" are exactly the modules over $S[t^\pm1]$, the free associative ring spectrum on one invertible generator in degree 1. Dunno how helpful this observation is $\endgroup$ – Denis Nardin Aug 10 at 8:46
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    $\begingroup$ This doesn't help you to get a classification, but a simple way to construct a 1-periodic spectra is to take a 2-periodic spectrum $X$ and take wedge with its suspension. $\endgroup$ – user43326 Aug 10 at 18:33

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