Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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.

share|improve this question
add comment

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.

share|improve this answer
    
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
1  
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
add comment

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.

share|improve this answer
    
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
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.