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Bousfield p-completion on spaces is a functor $(-)^{\wedge p}$ whose main property is that a map $f:X\rightarrow Y$ induces an isomorphism $f_{\ast}:H_\ast(X,\mathbb{F}_{p})\rightarrow H_\ast(Y,\mathbb{F}_p)$ if and only if $f^{\wedge p}:X^{\wedge p}\rightarrow Y^{\wedge p}$ is a homotopy equivalence.

Let $X$ be now be a connective CW spectra, it is known that $H\mathbb{F}_p$-localization agrees with $M\mathbb{F}_p$-localization (both on X, the latter is referred as the $p$-completion). Let $f:E\rightarrow F$ a map between two connective CW spectra, is there any connection between $f^{\wedge p}$ and $({H\mathbb{F}_p})_{\ast}(f)$ (or $({H\mathbb{F}_p})^{\ast}(f)$)?, of course, an analogous version of the previous property is desirable, although there may be more restrictive conditions so that this property holds.

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    $\begingroup$ It can very well happen that the map is zero in homology, but is not nullhomotopic, e.g. almost any map between shifts of sphere spectra. In some sense, the more detailed relationship is described by the Adams spectral sequence. $\endgroup$
    – Lennart Meier
    Commented Jun 13, 2020 at 17:48
  • $\begingroup$ @LennartMeier Do you mean a map $f$ with $H_\ast(f,\mathbb{F}_p)=0$ but $f^{\wedge p}\not\simeq\ast$? $\endgroup$
    – Victor TC
    Commented Jun 13, 2020 at 21:49
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    $\begingroup$ Yes. This is fullfilled for any map $\Sigma^n\mathbb{S} \to \mathbb{S}$ with $n$ positive, whose order is a power of $p$, e.g. the Hopf map with $n=1$ and $p=2$. $\endgroup$
    – Lennart Meier
    Commented Jun 14, 2020 at 6:51
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    $\begingroup$ In view of the way the question is formulated, an obvious answer is $(H\mathbb{F}_p)_*(f)=(H\mathbb{F}_p)_*(f^{\wedge p})$, (and similarly for cohomology), is this what is really asked for? $\endgroup$
    – user43326
    Commented Jun 14, 2020 at 7:05
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    $\begingroup$ @VictorTC As a matter of fact it is more or less the definition of the localization (p-completion is the localization with respect to $HF_p$) $\endgroup$
    – user43326
    Commented Jun 14, 2020 at 18:36

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