BibSonomy publications for /user/dmartins/2-diskSun Mar 02 02:12:02 CET 2008161-175The space of simplexwise linear homeomorphisms of a convex 2-disk.231984space of homotopy the groups linear homeomorphisms a 2-disk convex simplexwise Let $K\sp n$ be a finite simplicial complex whose underlying space $\vert K\sp n\vert$ is a combinatorial n-dimensional disk in ${\bbfR}\sp n$. Let $L(K\sp n)$ be the space, with the compact open topology, of all the homeomorphisms of $\vert K\sp n\vert$ that are affinely linear on each simplex of $K\sp n$ and the identity on B$d(\vert K\sp n\vert)$. 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 {\it S. S. Cairns}, Ann. Math., II. Ser. 45, 207-217 (1944); {\it R. Thom}, Proc. Int. Congr. Math. 1958, 248-255 (1960; Zbl 137, 426) and {\it 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 $S\sp 3$ is of the same homotopy type as the special orthogonal group SO(4) [see {\it 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 $L(K\sp n)$ is homeomorphic to the Euclidean space ${\bbfR}\sp{2k}$, where k is the number of interior vertices (i.e., vertices not lying on B$d(\vert K\sp n\vert))$ of $K\sp n$. 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 $\pi\sb 0(L(K\sp 2))=0$ and of the reviewer that $\pi\sb 1(L(K\sp 2))=0$. 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.