What is the homotopy type of the group of diffeomorphisms of the 4-sphere?

Importance: Outstanding ✭✭✭✭
Author(s): Smale, S.
Subject: Topology
Recomm. for undergrads: no
Posted by: rybu
on: November 7th, 2009

\begin{problem} $Diff(S^4)$ has the homotopy-type of a product space $Diff(S^4) \simeq \mathbb O_5 \times Diff(D^4)$ where $Diff(D^4)$ is the group of diffeomorphisms of the 4-ball which restrict to the identity on the boundary. Determine some (any?) homotopy or homology groups of $Diff(D^4)$. \end{problem}

$Diff(D^4$) is known to be a $5$-fold loop space. In particular there is a homotopy-equivalence known as the Cerf-Morlet Comparison theorem $Diff(D^n) \simeq \Omega^{n+1} (PL_n / O_n)$ where $PL_n$ is the group of PL-automorphisms of $\mathbb R^n$ and $O_n$ is the group of linear automorphisms of $\mathbb R^n$. Otherwise there is not much in the literature about $Diff(D^4)$. Since it is a group of diffeomorphisms it has the homotopy type of a countable CW-complex. It is unknown whether or not it is connected, or if it has any other non-trivial homotopy or homology groups.

$Diff(S^n)$ is known to have the homotopy-type of $O_{n+1}$ provided $n \leq 3$ by work of Hatcher and Smale respectively. For $n \geq 5$ many of the groups $\pi_0 Diff(S^n)$ were computed by Kervaire and Milnor, who further related these groups to the homotopy groups of spheres. For $n \geq 7$ the rational homotopy groups of $Diff(D^n)$ have been computed by Farrell and Hsiang in range $0 \leq i < \min\{\frac{n-4}{3}, \frac{n-7}{2} \}$. They show $\pi_i Diff(D^n) \otimes \mathbb Q \simeq \left\{ \begin{array}{lr} \mathbb Q & \text{ provided }\ 4 | (i+1) \\ 0 & \text{ otherwise } \end{array} \right.$. % You may use many features of TeX, such as % arbitrary math (between $...$ and $$...$$) % \begin{theorem}...\end{theorem} environment, also works for question, problem, conjecture, ... % % Our special features: % Links to wikipedia: \Def {mathematics} or \Def[coloring]{Graph_coloring} % General web links: \href [The On-Line Encyclopedia of Integer Sequences]{http://www.research.att.com/~njas/sequences/}


[B] Budney, R. Little cubes and long knots. Topology. 46 (2007) 1--27.

[FH] Farrell, F.T. Hsiang, W.C. On the rational homotopy groups of the diffeomorphism groups of discs, spheres and aspherical manifolds. Proc. Symp. Pure. Math. 32 (1977) 403--415.

[H] Hatcher, A proof of a Smale conjecture, ${\rm Diff}(S\sp{3})\simeq {\rm O}(4)$. Ann. of Math. (2) 117 (1983), no. 3, 553--607.

[KS] Kirby, R. Siebenmann, L. Foundational Essays on Topological Manifolds, Smoothings, and Triangulations. Princeton University Press.

*[S] Smale, S. Diffeomorphisms of the 2-sphere, Proc. Amer. Math. Soc. 10 (1959) 621--626.

* indicates original appearance(s) of problem.