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There is an inclusion $S_n \to Aut(F_n)$ from the symmetric group into the automorphism group of a free group. After applying the Quillen +-constriction, both $BS_{\infty}$ and $BF_{\infty}$ become infinite loop spaces and the map above can be promoted to an infinite loop space map $BS_{\infty}^+ \to BF_{\infty}^+$.

Are there any other infinite loop space maps into or our of $Aut(F_{\infty})?$

Some notes:

1) Galatius has shown relatively recently that $BAut(F_{\infty})^+$ is homotopic to the sphere spectrum; the latter has been known for about 4 decades to be homotopic to $BS_{\infty}^+$ (Barratt-Priddy-Segal). So loop maps in or out of the symmetric group can all be defined up to homotopy on $Aut(F_{\infty})$ instead. Also, in all of the above, can "Aut" be replaced by "Out"?

2) Answers involving $\Omega^{\infty}S^{\infty}$ that don't relate to spaces of graphs or free groups don't count..count - I'm trying to get information on this from a geometric group theory point of view.

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There is an inclusion $S_n \to Aut(F_n)$ from the symmetric group into the automorphism group of a free group. After applying the Quillen +-constriction, both $BS_{\infty}$ and $BF_{\infty}$ become infinite loop spaces and the map above can be promoted to an infinite loop space map $BS_{\infty}^+ \to BF_{\infty}^+$.

Are there any other infinite loop space maps into or our of $Aut(F_{\infty})?$

Some notes:

1) Galatius has shown relatively recently that $BAut(F_{\infty})^+$ is homotopic to the sphere spectrum; the latter has been known for about 4 decades to be homotopic to $BS_{\infty}^+$ (Barratt-Priddy-Segal). So loop maps in or out of the symmetric group can all be defined up to homotopy on $Aut(F_{\infty})$ instead. Also, in all of the above, can "Aut" be replaced by "Out"?

2) Answers involving $\Omega^{\infty}S^{\infty}$ that don't relate to spaces of graphs or free groups don't count...

2 edited title; added 34 characters in body

# ArethereanyotherinfiniteInfinite loop space maps into or out of BAut(F_n)?BAut(F_n)

There is an inclusion $S_n \to Aut(F_n)$ from the symmetric group into the automorphism group of a free group. After applying the Quillen +-constriction, both $BS_{\infty}$ and $BF_{\infty}$ become infinite loop spaces and the map above can be promoted to an infinite loop space map $BS_{\infty}^+ \to BF_{\infty}^+$.

Are there any other infinite loop space maps into or our of $Aut(F_{\infty})?$

Some notes: Galatius has shown relatively recently that $BAut(F_{\infty})^+$ is homotopic to the sphere spectrum; the latter has been known for about 4 decades to be homotopic to $BS_{\infty}^+$ (Barratt-Priddy-Segal). So loop maps in or out of the symmetric group can all be defined up to homotopy on $Aut(F_{\infty})$ instead. Also, in all of the above, can "Aut" be replaced by "Out"?

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