Homotopy type of the self-homotopy equivalences of a bouquet of spheres

Before I state the questions I have in mind, let me give some background. If one considers $S^2$ then it is known due to Kneser that $\textrm{Homeo}^{+}(S^2)$ has the homotopy type of $SO(3)$. By Smale's work we also know that $\textrm{Diff}^{+}(S^2)$ is homotopic to $SO(3)$. However, when we work in the homotopy category this changes. Later Hansen considered $\textrm{Aut}_0(S^2)$, the connected component of identity in the space of all self homotopy equivalences. He showed that its homotopy type is that of $SO(3)\times \mathbf{\Omega}$, where $\mathbf{\Omega}$ is the universal cover of the connected component of the constant loop in the double loop space $\Omega^2 S^2$.

Although this is a very nice and interesting fact, it's turns out to be hard to generalize his method of proof for higher spheres. For one, the homotopy type of $\textrm{Diff}(S^n)$ is not really known for $n\geq 7$ if I'm not mistaken. For another, Hansen crucially uses the fact that $S^2$ is the base of the usual fibration $SO(2)\to SO(3)\to S^2$ and the fact $SO(2)=S^1$ has no higher homotopy. This indubitably fails for higher spheres!

Question 1 What is known about the homotopy type of $\textrm{Aut}_0(S^n)$ for $n\geq 3$?

I should say that the rational homotopy type is fairly easily calculable via Sullivan's minimal models. So, I'm looking for a bit more here.

Question 2 What is the homotopy type of the identity component of self homotopy equivalences of $\vee_k S^2$, a bouquet of $2$-spheres?

Of course, one can ask this question for higher spheres but bearing in mind question 1, I decided I would be happy with an answer for $S^2$. If it helps, the homotopy groups of $\vee_k S^2$ can be calculated by Hilton-Milnor theorem. This fits in a long exact sequence of groups associated to $\textrm{Aut}_0^\ast(\vee_k S^2)\to \textrm{Aut}_0(\vee_k S^2)\to \vee_k S^2$ where the last map is evaluation of an automorphism at the common point of the bouquet and $\textrm{Aut}^\ast_0(\vee_k S^2)$ consists of based maps. But this doesn't seem to lead anywhere!

• Do you have any idea what would happen in the case of completing with respect to a prime? You seem to understand the rational information. Maybe you can put it all together to reasemble the homotopy type from the prime pieces and the rational piece. Nov 23 '11 at 4:03
• @Spice the Bird - You may be onto something here! Unfortunately, I have no idea of what the theory at a prime p is although I've heard people using these words. Do you have any readable reference (of the general theory and then reassembling rational and prime pieces) in mind? I can perhaps try to learn something new at least! Nov 23 '11 at 4:16
• You might take a look at the paper of Dror-Dwyer-Kan: nd.edu/~wgd/Html/Arithmetic.Square.html Or Sullivan's older papers (either the Annals paper referenced by the paper above or his "MIT notes" on geometric topology, which were finally published a few years ago: books.google.com/books/about/…). Nov 23 '11 at 8:12
• I learned of the idea from Sullivan's MIT notes. Nov 23 '11 at 11:19
• I'm sure you realize this but Question 2 is also amenable to rational homotopy theory treatment using Haefliger construction of models for $Map_f(X,Y)$. Also, given a minimal model $(M,d)$ for $X$ the differential graded lie algebra of derivations of $M$ (with the differential given by $[-,d]$) is the Quillen Lie model for $BAut_0(X)$. Note however, that even for a wedge of two spheres (of any dimension) you'll get a rationally hyperbolic space and rational homotopy and homology of $Aut_0(X)$ will grow exponentially. So there won't be any nice answers here as there were for $Aut(S^n)$. Nov 23 '11 at 15:33

For a reasonably nice space $X$ (say, a finite connected CW-complex) let $Aut_0(X)$ be the monoid of self-equivalences of $X$ homotopic to the identity and let $Aut_0^\bullet(X)$ be the submonoid fixing a basepoint. Then there is a natural fibration given by the evaluation map $Aut_0^\bullet(X)\to Aut_0(X)\to X$. This gives rise to the fibration $X\to BAut_0^\bullet(X)\to BAut_0(X)$ which is the universal fibration with fiber $X$.
For $X=S^n$ it's easy to see that $Aut_0^\bullet(S^n)\cong Map_0^\bullet(S^n, S^n)\cong Map_0^\bullet(S^{n-1}, \Omega S^n)\cong \ldots Map_0^\bullet (S^0, \Omega^n(S^n))=\Omega^n(S^n)$. In particular, $\pi_k(Aut_0^\bullet(S^n))\cong \pi_{n+k}(S^n)$. Combining this with the fibration $Aut_0^\bullet(S^n)\to Aut_0(S^n)\to S^n$ this in principle gives you homotopy groups of $Aut_0(S^n)$. In practice though I don't think this is very useful away from rational coefficients or for small values of $k$ where you have stability.
I'm not familiar with the general theory for spaces other than spheres. As I mentioned in my comment, rationally the situation is in principle well understood. But even for rational coefficients the homotopy structure of $Aut_0(X)$ is complicated when $X$ is a wedge of spheres. As I mentioned, rational homotopy groups of $Aut_0(X)$ grow exponentially. See "The monoid of self-homotopy equivalences of some homogeneous spaces" by Félix and Thomas for a careful proof of that. One general fact that I know is a result of Dror, Dwyer and Kan that when $X$ is a nilpotent finite $CW$-complex then $BAut_0(X)$ has finite type, i.e. it has the homotopy type of a CW complex with a finite number of cells in each dimension.