Article,
The space of simplexwise linear homeomorphisms of a convex 2-disk.
(1984)
Abstract
Let $Kn$ be a finite simplicial complex whose underlying space
$Kn\vert$ is a combinatorial n-dimensional disk in $\bbfR\sp
n$. Let $L(Kn)$ be the space, with the compact open topology,
of all the homeomorphisms of $Kn\vert$ that are affinely
linear on each simplex of $Kn$ and the identity on B$d(\vert
Kn\vert)$. Interest in the homotopy properties of the space $L(K\sp
n)$ was first initiated with the smoothing theory. Conditions on
the existence and uniqueness of differentiable structures on a combinatorial
manifold can be formulated in terms of the homotopy groups of this
and some related spaces see S. S. Cairns, Ann. Math., II.
Ser. 45, 207-217 (1944); R. Thom, Proc. Int. Congr. Math. 1958,
248-255 (1960; Zbl 137, 426) and N. H. Kuiper, Diff. and Comb.
Topology, 3-22 (1965; Zbl 171, 444). Recently, there has been a
revival of interest in these spaces due to their connection to the
Smale conjecture that the space of all the orientation preserving
diffeomorphisms of $S3$ is of the same homotopy type as the special
orthogonal group SO(4) see A. E. Hatcher, Proc. Int. Congr.
Math., Helsinki 1978, Vol. 2, 463-468 (1980; Zbl 455.57014). \par
In the present paper, the authors prove that when $n=2$, the space
$L(Kn)$ is homeomorphic to the Euclidean space $\bbfR2k$,
where k is the number of interior vertices (i.e., vertices not lying
on B$d(Kn\vert))$ of $Kn$. This is about the best possible
result one could obtain for the two-dimensional case and is a great
improvement of the previous results of Cairns that $\pi0(L(K\sp
2))=0$ and of the reviewer that $\pi1(L(K2))=0$. As a consequence
of the present result, the following theorem of Smale can be derived
as a corollary: the space of diffeomorphisms of a smooth 2-disk,
fixed on the boundary, is contractible. The proof of the present
result involves some difficult and ingenious geometric arguments.
