Timeline for Infinite loop of a p-completed specta vs p-completion of infinite loop of the spectra

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Apr 6, 2014 at 16:23	comment	added	user43326		You are right, I added $0$-connected hypothesis to avoid the $\pi _0$ issue. As to the non-finiteness, I don't know if there is any good theory to deal with this, but anyway we still have the Milnor exact sequence, so we can still compute the homotopy groups of $\Omega ^{\infty}(X_p^{\wedge})$.
Apr 6, 2014 at 16:19	history	edited	user43326	CC BY-SA 3.0	Added $0$-connected assumption to avoid the $\pi _0$ issue
Apr 6, 2014 at 0:03	comment	added	Sam Nariman		I think there is $\pi_0$ issue, even for $X=H\mathbb{Z}$ we don't have $\Omega^{\infty}X^{\wedge}_p\simeq (\Omega^{\infty}X)^{\wedge}_p$ because they don't have the same $\pi_0$. That is why I asked for $\Omega^{\infty}_0X^{\wedge}_p\simeq (\Omega^{\infty}_0X)^{\wedge}_p$. The thing is I don't want to impose finite type conditions because I am working with Thom spectrum over $BG^{\delta}$ where $G^{\delta}$ is some lie group with discrete topology. It is connective spectrum but not of finite type.
Apr 3, 2014 at 8:16	history	answered	user43326	CC BY-SA 3.0