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This question is probably trivial for anyone who is more familiar with rational homotopy theory than me, but anyway:

Let $G$ be a simply-connected topological group. In particular, it is an $H$-space. Therefore the theorem of Cartan and Serre tells us that the rationalization $G_{\mathbb{Q}}$ of $G$ has the homotopy type of a product of Eilenberg-Maclane spaces. Here are my questions:

  • Can one choose a functorial rationalization $G \to G_{\mathbb{Q}}$ that also preserves products, thereby turning $G_{\mathbb{Q}}$ into a topological group and $G \to G_{\mathbb{Q}}$ into a group homomorphism? Would this group structure coincide up to homotopy with the $H$-space structure on the product of EM-spaces?
  • If the above is true, then the map $(BG)_{\mathbb{Q}}\to B(G_{\mathbb{Q}})$ induced by $G \to G_{\mathbb{Q}}$ should be a homotopy equivalence. Would this imply that $(BG)_{\mathbb{Q}}$ also decomposes into a product of Eilenberg-Maclane spaces?
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  • $\begingroup$ @QiaochuYuan: Is it true that you can realize a class by a group homomorphism $G \to K(\pi_n(G) \otimes \mathbb{Q},n)$? $\endgroup$ – Ulrich Pennig Apr 1 '14 at 5:01
  • $\begingroup$ Hi Uli! A finite-dimensional Lie group is rationally a product of odd spheres, so I wouldn't expect a reasonable group structure on $G_\mathbb{Q}$ (although this is far from being a no-go argument)... $\endgroup$ – Christoph Wockel Apr 1 '14 at 18:10
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In the paper

Arkowitz, Martin Categories equivalent to the category of rational H-spaces, Manuscripta Math. 64 (1989), no. 4, 419–429

it is shown that the rational homotopy equivalence $G_\mathbb{Q}\cong\prod_n K(\pi_n(G_\mathbb{Q}),n)$ is an equivalence in the category of rational $H$-spaces if and only if $G$ is homotopy abelian (see Proposition 3.1). This means, I think, that the answer to your first question is no in general.

However I think the answer to your second question should be yes, but for different reasons. If $G$ is a connected topological group of the homotopy type of a finite CW complex, then theorems of Hopf and Borel imply that $$H^\ast(BG;\mathbb{Q})\cong\mathbb{Q}[y_1,\ldots , y_k]$$ where each generator $y_i$ is of even degree (see McCleary's A User's Guide to Spectral Sequences, Theorem 6.38). It follows that the minimal model of $BG$ is evenly generated, and has zero differential, meaning that $BG$ is rationally a product of Eilenberg-Mac Lane spaces.

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  • $\begingroup$ Suppose I have a group, such that $H^*(BG; \mathbb{Q}) \cong \mathbb{Q}[y_1, y_2, \dots]$ with infinitely many $y_i$'s in even degrees. Would this argument still work? $\endgroup$ – Ulrich Pennig Apr 4 '14 at 19:57
  • $\begingroup$ Two nilpotent spaces with finite Betti numbers are rationally homotopy equivalent if and only if they have isomorphic minimal models. So the above argument should work, as long as there are only finitely many $y_i$ in each even degree. $\endgroup$ – Mark Grant Apr 5 '14 at 9:15
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    $\begingroup$ Commutativity shows that the answer to the second question in the first bullet point is "No," but does not address the first question in that bullet point. I think your second argument actually shows that finite dimensional rational $H$-spaces are commutative, so it really is the same reason. An infinite dimensional non-commutative example is given by $\Omega S^{2n}$. Its classifying space $S^{2n}$ is not an $H$-space, let alone a product of EM spaces, not even after rationalization. $\endgroup$ – Ben Wieland Apr 5 '14 at 20:29
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This is a longish comment on James Schwass's answer, not an answer to the original questions.

Have to be a little careful here. We are deliberately informal (p.97), but we are working in a category of "cocellular" spaces and maps on which localizations can be constructed to be product-preserving on the point-set level. Every nilpotent space $X$ "admits" such a Postnikov tower $Y$, but that only gives you a weak equivalence $X\to Y$. Our arguments give a more precise than usual version of the fact that rationalization commutes with products up to homotopy: cocellular spaces admit cocellular localizations; products and more generally pullbacks of cocellular localizations are cocellular localizations. Incidentally, Bousfield and Kan have nothing to do with this. On nilpotent spaces, localizations are the same up to homotopy no matter what construction one uses. Bousfield-Kan gives a particular simplicial construction on the point-set level, but that construction is not used in May-Ponto. Note that on non-nilpotent spaces there are several different notions of localization (and in particular rationalization), none well understood calculationally.

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  • $\begingroup$ Ah, it seems I've got much more to learn about localization. Thanks for setting the record straight! $\endgroup$ – James Schwass Apr 8 '14 at 15:51

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