Let $X$ be a commutative H-space. A group completion is an H-map $X\to Y$, where $Y$ is another H-space, such that

• $\pi_0(Y)$ is a group
• The Pontrjagin ring $H(Y; R)$ is the localization of the Pontrjagin ring $H_*(X; R)$ at the multiplicative submonoid $\pi_0(X)$ for every coefficient ring $R$.

Perhaps most interesting is the case where $X$ is a commutative monoid or more generally an $E_\infty$-space. In this case, May gives a functorial group completion $B_0$ via the two-sided bar construction which is an infinite-loop space. If I understand it right, it is defined as $colim_j \Omega^j|\Sigma^j(C_j\times C')^\bullet X|$, where $C_j\times C'$ denotes the monad associated to the product of the little $j$-cube operad and an $E_\infty$-operad and $\bullet$ the simplicial variable. I have some question concerning group completions:

1. Are all group completions equivalent? That is, does there always exist a homotopy equivalence of H-spaces between them?

2. Preserve group completions homotopy limits in some sense? For example, preserves the May functor homotopy limits of $E_\infty$-spaces?

3. Suppose one knows that all loop spaces of $X$ are infinite loop spaces. Is there a simple relationship between the infinite-loop space $B_0 X$ and the loop spaces of $X$? Especially, I am interested in the homotopy groups of $B_0 X$.

An answer to any of these question would be helpful to me.

Let $X$ be a commutative H-space. A group completion is an H-map $X\to Y$, where $Y$ is another H-space, such that

• $\pi_0(Y)$ is a group
• The Pontrjagin ring $H(Y; R)$ is the localization of the Pontrjagin ring $H_*(X; R)$ at the multiplicative submonoid $\pi_0(X)$ for every coefficient ring $R$.

Perhaps most interesting is the case where $X$ is a commutative monoid or more generally an $E_\infty$-space. In this case, May gives a functorial group completion $B_0$ via the two-sided bar construction which is an infinite-loop space. If I understand it right, it is defined as $colim_j \Omega^j|\Sigma^j(C_j\times C')^\bullet X|$, where $C_j\times C'$ denotes the monad associated to the product of the little $j$-cube operad and an $E_\infty$-operad and $\bullet$ the simplicial variable. I have some question concerning group completions:

1. Are all group completions equivalent? That is, does there always exist a homotopy equivalence of H-spaces between them?

2. Does the group completion preserve homotopy limits? For example, does the May functor preserve homotopy limits of $E_\infty$-spaces?

3. Suppose one knows that all loop spaces of $X$ are infinite loop spaces. Is there a simple relationship between the infinite-loop space $B_0 X$ and the loop spaces of $X$? Especially, I am interested in the homotopy groups of $B_0 X$.

An answer to any of these question would be helpful to me.

Let $X$ be a commutative H-space. A group completion is an H-map $X\to Y$, where $Y$ is another H-space, such that

• $\pi_0(Y)$ is a group
• The Pontrjagin ring $H(Y; R)$ is the localization of the Pontrjagin ring $H_*(X; R)$ at the multiplicative submonoid $\pi_0(X)$ for every coefficient ring $R$.

Perhaps most interesting is the case where $X$ is a commutative monoid or more generally an $E_\infty$-space. In this case, May exhibits a functorial group completion $B_0$ via the two-sided bar construction which is an infinite-loop space. If I understand it right, it is defined as $colim_j \Omega^j|\Sigma^j(C_j\times C')^\bullet X|$, where $C_j\times C'$ denotes the monad associated to the product of the little $j$-cube operad and an $E_\infty$-operad and $\bullet$ the simplicial variable. I have some question concerning group completions:

1. Are all group completions equivalent?

2. Preserve group completions homotopy limits in some sense? For example, preserves the May functor homotopy limits of $E_\infty$-spaces?

3. Suppose one knows that all loop spaces of $X$ are infinite loop spaces. Is there a simple relationship between the infinite-loop space $B_0 X$ and the loop spaces of $X$? Especially, I am interested in the homotopy groups of $B_0 X$.

Let $X$ be a commutative H-space. A group completion is an H-map $X\to Y$, where $Y$ is another H-space, such that

• $\pi_0(Y)$ is a group
• The Pontrjagin ring $H(Y; R)$ is the localization of the Pontrjagin ring $H_*(X; R)$ at the multiplicative submonoid $\pi_0(X)$ for every coefficient ring $R$.

Perhaps most interesting is the case where $X$ is a commutative monoid or more generally an $E_\infty$-space. In this case, May gives a functorial group completion $B_0$ via the two-sided bar construction which is an infinite-loop space. If I understand it right, it is defined as $colim_j \Omega^j|\Sigma^j(C_j\times C')^\bullet X|$, where $C_j\times C'$ denotes the monad associated to the product of the little $j$-cube operad and an $E_\infty$-operad and $\bullet$ the simplicial variable. I have some question concerning group completions:

1. Are all group completions equivalent? That is, does there always exist a homotopy equivalence of H-spaces between them?

2. Preserve group completions homotopy limits in some sense? For example, preserves the May functor homotopy limits of $E_\infty$-spaces?

3. Suppose one knows that all loop spaces of $X$ are infinite loop spaces. Is there a simple relationship between the infinite-loop space $B_0 X$ and the loop spaces of $X$? Especially, I am interested in the homotopy groups of $B_0 X$.

An answer to any of these question would be helpful to me.