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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 - I'm trying to get information on this from a geometric group theory point of view.

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2 Answers

Let $Aut(\bigvee^k S^n)$ be the topological monoid of pointed homotopy equivalences from a wedge sum of $k$ copies of $S^n$ to itself. Then $Aut(\bigvee^k S^0) = S_k = \Sigma_k$. The fundamental group functor induces a homotopy equivalence $Aut(\bigvee^k S^1) \simeq Aut(F_k)$. The $n$-th homology functor induces a homomorphism $Aut(\bigvee^k S^n) \to Aut(Z^k) = GL_k(Z)$. Suspension induced maps $$ B\Sigma_k \to BAut(F_k) \to \dots \to BAut(\bigvee^k S^n) \to \dots \to BGL_k(S) \to BGL_k(Z) . $$ Here $S$ is the sphere spectrum. Taking the wedge sum with the identity map on $S^n$ allows $k$ to grow. In the colimit we get $$ BS_\infty \to BAut(F_\infty) \to \dots \to BAut(\bigvee^\infty S^n) \to \dots \to BGL_\infty(S) \to BGL_\infty(Z) . $$ Taking the plus-construction and multiplying with $Z$ you get infinite loop maps $$ Q(S^0) \to Z \times BAut(F_\infty)^+ \to \dots \to A(\ast) \to K(Z) $$ where $A(\ast)$ is Waldhausen's algebraic $K$-theory of a point. The intermediate infinite loop spaces $Z \times BAut(\bigvee^\infty S^n)^+$ for $1 < n < \infty$ seem to be poorly understood.

An early reference:

Friedhelm Waldhausen, Algebraic $K$-theory of topological spaces. II. Algebraic topology, Aarhus 1978 (Proc. Sympos., Univ. Aarhus, Aarhus, 1978), pp. 356–394, Lecture Notes in Math., 763, Springer, Berlin, 1979.

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Ah, this is great, thanks. $Q(S^0) = Z \times BS_{\infty}^+ \to A(\ast) = Z \times BGL(S)^+$ is basically (finite set with n elements) $\to$ (n-frame) on the second factor. There's a clear map $BAut(S^{\infty}) \to A(\ast)$, basically inclusion into $BGL_{\infty}$ if I'm getting this right (followed by +-construction and crossign with Z). Looking at the map $BO \to BAut(S^{\infty})$, is there a map $BO \to Q(S^0)$ making a nice square? –  Romeo Oct 22 '10 at 4:42
I don't quite understand your notation, but, yes, there is a known map $BO \to Q(S^0)$ so that the composite $BO \to Q(S^0) \to A(\ast)$ is homotopic to the composite $BO \to BGL_1(S) \to A(\ast)$, where $BO \to BGL_1(S)$ is the $j$-map. With multiplicative infinite loop structures on the unit components of $Q(S^0)$ and $A(\ast)$ these are even infinite loop maps. Refs: F. Waldhausen, Algebraic $K$K-theory of spaces, a manifold approach, MR0686115 J. Rognes, The Hatcher-Waldhausen map is a spectrum map, MR1282230. –  John Rognes Oct 22 '10 at 14:36
Will read those papers, looks like they may be just what I want. Sorry, I wrote $Aut(S^{\infty})$ for some reason; I just meant lim $n \to \infty deg +/- 1 self maps of S^n$, which I I'll call $G$. The map between homotopy fibers G/O $\to$ Unknown Space must be interesting. But I'll hold off any speculations until I read the references you gave. –  Romeo Oct 22 '10 at 15:11
Yes, the homotopy fiber of $Q(S^0) \to A(\ast)$, that you call "Unknown Space", is $\Omega Wh^{Diff}(\ast)$, the stable smooth $h$-cobordism space of a point. There is map to it from the space $H^{Diff}(D^n)$ of $h$-cobordisms on $D^n$, and the connectivity of the map grows to infinity with $n$. The map $G/O \to \Omega Wh^{Diff}(\ast)$ is a rational equivalence, and is precisely $8$-connected after $2$-completion. Refs: M. Bökstedt, The rational homotopy type of $\Omega Wh^{Diff}(\ast)$, MR0764574, and J. Rognes, Two-primary algebraic $K$K-theory of pointed spaces, MR1923990. –  John Rognes Oct 22 '10 at 15:48
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In the following paper

U. Tillmann, The representation of the mapping class group of a surface on its fundamental group in stable homology, Q J Math (2010) 61 (3): 373-380.

Ulrike Tillmann studies the effect of the homomorphism $\Gamma_{g,1} \to \mathrm{Aut}(F_{2g})$ from the mapping class group of a surface of genus $g$ with one boudary, that sends a mapping class to its effect on the fundamental group of the surface. She also studies many variants of this. The main technical theorem is that $$\mathbb{Z} \times B\Gamma_\infty^+ \to \mathbb{Z} \times B\mathrm{Aut}(F_\infty)^+$$ is a map of infinite loop spaces, that it is a split epimorphism at odd primes, and that at 2 there is a slightly more complicated but still understood behavior.

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Wasn't aware of that paper, looks very interesting - will read very soon, thanks. Had only seen her papers on the arxiv.... –  Romeo Oct 22 '10 at 1:47
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