A paper by Mcduff and Segal justifies the following definition: A map of h-spaces $X \to Y$ is a group completion if the map is a localization on homology.

In the paper they prove that when $X$ is a topological monoid, the canonical map $X \to \Omega B X$ is a group completion in the above sense.

Unfortunately the example where $X=\sqcup_n B\Sigma_n $ shows that neither the space $Y$, nor the map, is unique up to homotopy:

Here the map $X \to Y=\mathbb{Z} \times B \Sigma_\infty$ is a group completion. So is the map $X \to Y_+=\mathbb{Z} \times (B \Sigma_\infty)_+$. $Y$ has homotopy groups that are concentrated in degree 1, whereas $Y_+$ has as its homotopy groups the stable homotopy groups of spheres.

In this case the map $X \to Y \to Y_+$ and $X \to Y_+$ are the same up to homotopy. This raises the question

Question: Given an H-space $X$, is any group completion $X \to Y$ followed by the plus construction map going to be unique up to homotopy?

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    $\begingroup$ $\mathbb Z× B\Sigma_\infty$ is not an H-space. $\endgroup$ Commented Jan 19, 2019 at 7:37
  • $\begingroup$ Yes it is - its a topological monoid. Think of it like a baby topological k-theory version of $\mathbb{Z} \times BU$. There's a map $\Sigma_n \times \Sigma_m\to \Sigma_{nm}$ which induces $B \Sigma_\infty \times B \Sigma_\infty \to B \Sigma_\infty$ coming from taking tensor product of matrices. $\mathbb{Z} \times B \Sigma_\infty$ has a monoid structure as the direct product of monoids. Notice also that the shift maps are inverted in this monoid, and hence $\mathbb{Z} \times B \Sigma_\infty$ is localized by an element in each connected component. $\endgroup$ Commented Jan 19, 2019 at 15:51
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    $\begingroup$ An H-space has abelian $\pi_1$. In particular H-spaces are not affected by the plus construction. $\endgroup$ Commented Jan 19, 2019 at 17:02
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    $\begingroup$ In general I recommend as a modern, though a bit more advanced, source for group completion Thomas Nikolaus's notes: people.mpim-bonn.mpg.de/thoni/Papers/Group_completion.pdf $\endgroup$ Commented Jan 21, 2019 at 17:41
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    $\begingroup$ You could, additionally, also have a look at (the 6th chapter) of this paper by Ebert and Randal-Williams: arxiv.org/pdf/1705.03774.pdf $\endgroup$ Commented Jan 24, 2019 at 14:18


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