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EDIT: In general, for a path connected space $X$ the stable James-Hopf invariants are not multiplicative which you can see by homology computations; look at Kuhn's paper `The Homology of the James-Hopf maps'. But, in the cases such as $X=S^0$ you may get a map which deloops once, but probably not an infinite loop map, even not an $\Omega^2$ map in the case of $Q_0S^0\to Q\mathbb{R}P$; again look at Kuhn's paper on the homology, Remark 2.15.

To elaborate this, either use May's little cube model $C$ or the $\Gamma^+$ of Barratt-Eccles for which they have published 3 papers on this model. The $\Gamma^+$ is defined over the category of simplicial spaces, and the relation to $C$ is that $|\Gamma^+X|$ is homotopy equivalent to $C|X|$ with $|\ |$ is the geometric realisation functor. The simplical space $\Gamma^+X$ is a monoid object whose groupl completion is denoted by $\Gamma X$. The inclusion $\Gamma^+X\to \Gamma X$ is a weak equivalence for $X$ path connected, and a group completion in general. For any simplicial space $X$, $|\Gamma X|$ is homotopy equivalent to $Q|X|$. There are two decomposition/splittings which one could be derived from the other one. First, a stable splitting, so called Snaith splitting, $$\Sigma^\infty\Gamma^+ X\simeq\bigvee \Sigma^\infty D_rX$$ which upon projection to the $r$-th factor, after taking stable adjoint yields the $r$-the James-Hopf map $\Gamma^+X\to \Gamma D_rX$. Second is Milnor type splitting $$\Gamma\Gamma^+ X\to \prod \Gamma D_rX.$$

To second from the first you may apply $\Omega^\infty$ noting $\Omega^\infty$ takes sums to products.

Now, take the first splitting, project onto the $r$-factor and the take stable adjoint. That gives $j_r:\Gamma^+X\to\Gamma D_rX$ known as the stable $r$-th James-Hopf map; when $X$ is path connected, this gives the famous James-Hopf invariants $QX\to QD_rX$. But, when $X$ is not path connected, the situation is a little bit different. Set $D_0X=S^0$ and define $j_0:\Gamma^+X\to \Gamma D_0X$ by sending everything to $1$. The maps $j_r$ for $j\geq 0$ then piece together to yield a map $$j:\Gamma^+X\to\prod \Gamma D_rX\simeq \Gamma(\bigvee D_rX).$$ The target is a ring space, one coming from $\Gamma$ and one coming from the pairings $D_rX\wedge D_sX\to D_{r+s}X$. This map is known to have exponential property, sending the additive structure on $\Gamma^+$ to the one coming from the outer $\Gamma$, and in particular a map of monoids. Since, the group completion of a group is it self, then applying the group completion $\Omega B$ to the map $j$ and projecting onto the $r$-factor gives a map which may deloop once, because of the $\Omega$ in the group completion $\Omega B$.

Noting that $D_rS^0={B\Sigma_r}_+$, I think you can now derive a map $Q_0S^0\to Q {B\Sigma_r}_+$ which deloops once (because of the $\Omega$ in the group completion).

EDIT: In general, for a path connected space $X$ the stable James-Hopf invariants are not multiplicative which you can see by homology computations; look at Kuhn's paper `The Homology of the James-Hopf maps'. But, in the cases such as $X=S^0$ you may get a map which deloops once, but probably not an infinite loop map, even not an $\Omega^2$ map in the case of $Q_0S^0\to Q\mathbb{R}P$; again look at Kuhn's paper on the homology, Remark 2.15. In that paper, the author shows that there is a choice $Q_0S^0\to Q\mathbb{R}P$ which deloops once. You probably can do the same of other $B\Sigma_r$ factors.

You may look at papers of Barratt and Eccles, as well as Kuhn, to see how one may get James-Hopf maps $QX\to QD_rX$ when $X$ is not path connected.

To elaborate this, either use May's little cube model $C$ or the $\Gamma^+$ of Barratt-Eccles for which they have published 3 papers on this model. The $\Gamma^+$ is defined over the category of simplicial spaces, and the relation to $C$ is that $|\Gamma^+X|$ is homotopy equivalent to $C|X|$ with $|\ |$ is the geometric realisation functor. The simplical space $\Gamma^+X$ is a monoid object whose groupl completion is denoted by $\Gamma X$. The inclusion $\Gamma^+X\to \Gamma X$ is a weak equivalence for $X$ path connected, and a group completion in general. For any simplicial space $X$, $|\Gamma X|$ is homotopy equivalent to $Q|X|$. There are two decomposition/splittings. First, a stable splitting, so called Snaith splitting, $$\Sigma^\infty\Gamma^+ X\simeq\bigvee \Sigma^\infty D_rX$$ which upon projection to the $r$-th factor, after taking stable adjoint yields the $r$-the James-Hopf map $\Gamma^+X\to \Gamma D_rX$. Second is Milnor type splitting $$\Gamma\Gamma^+ X\to \prod \Gamma D_rX.$$

The two splittings are in fact the same, noting that you can apply $\Omega^\infty$ to the first and derive the second remembering that $\Omega^\infty$ takes sums to products.

Finally, lets take $\Omega B$ as model for group completion. Hence, $\Omega B(\Gamma^+X)$ is homotopy equivalent to $\Gamma X$.

To elaborate this, either use May's little cube model $C$ or the $\Gamma^+$ of Barratt-Eccles for which they have published 3 papers on this model. The $\Gamma^+$ is defined over the category of simplicial spaces, and the relation to $C$ is that $|\Gamma^+X|$ is homotopy equivalent to $C|X|$ with $|\ |$ is the geometric realisation functor. The simplical space $\Gamma^+X$ is a monoid object whose groupl completion is denoted by $\Gamma X$. The inclusion $\Gamma^+X\to \Gamma X$ is a weak equivalence for $X$ path connected, and a group completion in general. For any simplicial space $X$, $|\Gamma X|$ is homotopy equivalent to $Q|X|$. There are two decomposition/splittings which one could be derived from the other one. First, a stable splitting, so called Snaith splitting, $$\Sigma^\infty\Gamma^+ X\simeq\bigvee \Sigma^\infty D_rX$$ which upon projection to the $r$-th factor, after taking stable adjoint yields the $r$-the James-Hopf map $\Gamma^+X\to \Gamma D_rX$. Second is Milnor type splitting $$\Gamma\Gamma^+ X\to \prod \Gamma D_rX.$$

To second from the first you may apply $\Omega^\infty$ noting $\Omega^\infty$ takes sums to products.

Now, take the first splitting, project onto the $r$-factor and the take stable adjoint. That gives $j_r:\Gamma^+X\to\Gamma D_rX$ known as the stable $r$-th James-Hopf map; when $X$ is path connected, this gives the famous James-Hopf invariants $QX\to QD_rX$. But, when $X$ is not path connected, the situation is a little bit different. Set $D_0X=S^0$ and define $j_0:\Gamma^+X\to \Gamma D_0X$ by sending everything to $1$. The maps $j_r$ for $j\geq 0$ then piece together to yield a map $$j:\Gamma^+X\to\prod \Gamma D_rX\simeq \Gamma(\bigvee D_rX).$$ The target is a ring space, one coming from $\Gamma$ and one coming from the pairings $D_rX\wedge D_sX\to D_{r+s}X$. This map is known to have exponential property, sending the additive structure on $\Gamma^+$ to the one coming from the outer $\Gamma$, and in particular a map of monoids. Since, the group completion of a group is it self, then applying the group completion $\Omega B$ to the map $j$ and projecting onto the $r$-factor gives a map which may deloop once, because of the $\Omega$ in the group completion $\Omega B$.

For (1) Adams's book ``Infinite loop spaces'' contains good explanations. Also, a paper by Barratt and Priddy `On the Homology of Non-Connected Monoids and Their Associated Groups', I think is a good reference.

For (2), the maps you consider are the Stable James-Hopf invariants, and not Kahn-Priddy as far as I know, which arise from Snaith splitting. A good reference is Kuhn's paper on `The geometry of the James-Hopf maps'.

(3) and (4) are related to the extension of the Kahn-Priddy theorem (related to the case p=2 of Whitehead conjecture) to odd primes, and I think you can find proofs in Kuhn and Priddy's paper `The transfer and Whitehead's conjecture'

as well as more, what I call, `computation-free' work, in his other paper

`EXTENDED POWERS OF SPECTRA AND A GENERALIZED KAHN-PRIDDY THEOREM'.

Added, I think for the getting a $p$-local epimorphism $B\Sigma_p$ is not enough, and you need another $D_2$-factor for that. You may look at, Theorem 1.1(2) of the second paper for this extra factor.

For (1) Adams's book ``Infinite loop spaces'' contains good explanations. Also, a paper by Barratt and Priddy `On the Homology of Non-Connected Monoids and Their Associated Groups', I think is a good reference.

For (2), the maps you consider are the Stable James-Hopf invariants, and not Kahn-Priddy as far as I know, which arise from Snaith splitting. A good reference is Kuhn's paper on `The geometry of the James-Hopf maps'.

(3) and (4) are related to the extension of the Kahn-Priddy theorem (related to the case p=2 of Whitehead conjecture) to odd primes, and I think you can find proofs in Kuhn and Priddy's paper `The transfer and Whitehead's conjecture'

as well as more, what I call, `computation-free' work, in his other paper

`EXTENDED POWERS OF SPECTRA AND A GENERALIZED KAHN-PRIDDY THEOREM'.

Added, I think for the getting a $p$-local epimorphism $B\Sigma_p$ is not enough, and you need another $D_2$-factor for that. You may look at, Theorem 1.1(2) of the second paper for this extra factor.

EDIT: In general, for a path connected space $X$ the stable James-Hopf invariants are not multiplicative which you can see by homology computations; look at Kuhn's paper `The Homology of the James-Hopf maps'. But, in the cases such as $X=S^0$ you may get a map which deloops once, but probably not an infinite loop map, even not an $\Omega^2$ map in the case of $Q_0S^0\to Q\mathbb{R}P$; again look at Kuhn's paper on the homology, Remark 2.15.

To elaborate this, either use May's little cube model $C$ or the $\Gamma^+$ of Barratt-Eccles for which they have published 3 papers on this model. The $\Gamma^+$ is defined over the category of simplicial spaces, and the relation to $C$ is that $|\Gamma^+X|$ is homotopy equivalent to $C|X|$ with $|\ |$ is the geometric realisation functor. The simplical space $\Gamma^+X$ is a monoid object whose groupl completion is denoted by $\Gamma X$. The inclusion $\Gamma^+X\to \Gamma X$ is a weak equivalence for $X$ path connected, and a group completion in general. For any simplicial space $X$, $|\Gamma X|$ is homotopy equivalent to $Q|X|$. There are two decomposition/splittings. First, a stable splitting, so called Snaith splitting, $$\Sigma^\infty\Gamma^+ X\simeq\bigvee \Sigma^\infty D_rX$$ which upon projection to the $r$-th factor, after taking stable adjoint yields the $r$-the James-Hopf map $\Gamma^+X\to \Gamma D_rX$. Second is Milnor type splitting $$\Gamma\Gamma^+ X\to \prod \Gamma D_rX.$$

The two splittings are in fact the same, noting that you can apply $\Omega^\infty$ to the first and derive the second remembering that $\Omega^\infty$ takes sums to products.

Finally, lets take $\Omega B$ as model for group completion. Hence, $\Omega B(\Gamma^+X)$ is homotopy equivalent to $\Gamma X$.

Noting that $D_rS^0={B\Sigma_r}_+$, I think you can now derive a map $Q_0S^0\to Q {B\Sigma_r}_+$ which deloops once (because of the $\Omega$ in the group completion).