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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!

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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. –  Spice the Bird 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! –  Somnath Basu Nov 23 '11 at 4:16
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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/…). –  Dan Ramras Nov 23 '11 at 8:12
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I learned of the idea from Sullivan's MIT notes. –  Spice the Bird 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 $Aut(S^n)$. –  Vitali Kapovitch Nov 23 '11 at 15:33

1 Answer 1

Regarding question 1 the following is well known.

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.

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