The space of simplexwise linear homeomorphisms of a convex 2-disk
Let \(K^ n\) be a finite simplicial complex whose underlying space \(| K^ n|\) is a combinatorial n-dimensional disk in \({\mathbb{R}}^ n\). Let \(L(K^ n)\) be the space, with the compact open topology, of all the homeomorphisms of \(| K^ n|\) that are affinely linear on each simplex of \(K^ n\) and the identity on B\(d(| K^ n|)\). Interest in the homotopy properties of the space \(L(K^ n)\) 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 \textit{S. S. Cairns}, Ann. Math., II. Ser. 45, 207-217 (1944); \textit{R. Thom}, Proc. Int. Congr. Math. 1958, 248-255 (1960; Zbl 0137.426) and \textit{N. H. Kuiper}, Diff. and Comb. Topology, 3-22 (1965; Zbl 0171.444)]. 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 \(S^ 3\) is of the same homotopy type as the special orthogonal group SO(4) [see \textit{A. E. Hatcher}, Proc. Int. Congr. Math., Helsinki 1978, Vol. 2, 463-468 (1980; Zbl 0455.57014)]. In the present paper, the authors prove that when \(n=2\), the space \(L(K^ n)\) is homeomorphic to the Euclidean space \({\mathbb{R}}^{2k}\), where k is the number of interior vertices (i.e., vertices not lying on B\(d(| K^ n|))\) of \(K^ n\). 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 \(\pi_ 0(L(K^ 2))=0\) and of the reviewer that \(\pi_ 1(L(K^ 2))=0\). 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.
homotopy groups of the space of simplexwise linear homeomorphisms of a convex 2-disk
scientific article; zbMATH DE number 3872294