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Let $\mathrm{Top}$ be the model category of topological spaces. Define a new model structure on $\mathrm{Top}$ where $f:X\rightarrow Y$ is a weak equivalence iff $$f_{\ast}:H_{\ast}(X,\mathbb{F}_{p})\rightarrow H_{\ast}(Y,\mathbb{F}_{p})$$ is an isomorphism. Let $X \mapsto L(X)$ be a functorial fibrant replacement in the new model category.

Questions:

  1. If $X\rightarrow Y$ is an $E_{\infty}$-map between $E_{\infty}$-spaces, is it true that $L(X)\rightarrow L(Y)$ is an $E_{\infty}$-map between $E_{\infty}$-spaces?

  2. If $X$ is a pointed connected space (simply connected?), is it true that $$ L(\Omega X)\sim \Omega L(X) \, ? $$

  3. If $X$ is a connected $E_{\infty}$-space, is it true that $$ L(BX)\sim B(LX) $$ as $E_{\infty}$-spaces, where $B$ is the bar construction?

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    $\begingroup$ 1 is true because the product of two homology equivalences is a homology equivalence, so $L$ preserves finite products. $X=S^1$ is a counterexample to 2. $\endgroup$ – Marc Hoyois Jun 4 '15 at 17:44
  • $\begingroup$ @MarcHoyois thanks, so for 2 we need probably that X in simply connected at least... $\endgroup$ – Max Jun 4 '15 at 17:52
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I think all three are true (assuming $X$ simply connected in 2):

For 1, the key is the fact that the product of two homology equivalences is a homology equivalences (Künneth). So the map $L(X\times Y)\to LX\times LY$ is a homology equivalence, whence a weak equivalence since both sides are local. It follows that $L$ preserves finite homotopy products, and hence preserves all kinds of algebraic structures, including $E_\infty$-spaces.

For the last two, we will use that $L$ preserves connected spaces (because a homology equivalence induces an isomorphism on $\pi_0$) and that, for $X$ a nilpotent (e.g. with abelian $\pi_1$) pointed connected space and $n\geq 1$, there is a short exact sequence

$$ 0 \to \mathrm{Ext}(\mathbb{Z}_{p^\infty},\pi_n X) \to \pi_n LX \to \mathrm{Hom}(\mathbb{Z}_{p^\infty}, \pi_{n-1}X) \to 0. $$

(see for instance chapter VI of Bousfield and Kan, Homotopy limits, completions, and localizations).

It follows from this and the five lemma that, if $X$ is simply connected, the canonical map $L(\Omega X)\to \Omega (LX)$ is a weak equivalence. Similarly, if $X$ is a connected $E_\infty$-space (or even $E_1$), the canonical map $B(LX) \to LB(LX)\simeq L(BX)$ is a weak equivalence.

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