On certain homotopy properties of some spaces of linear and piecewise linear homeomorphisms. I

Abstract:

Let $K$ be a proper rectilinear triangulation of a $2$-simplex $S$ in the plane and $L(K)$ be the space of all homeomorphisms of $S$ which are linear on each simplex of $K$ and are fixed on $\text {Bd}(S)$. The author shows in this paper that $L(K)$ with the compact open topology is simply-connected. This is a generalization of a result of S. S. Cairns in 1944 that the space $L(K)$ is pathwise connected. Both results will be used in Part II of this paper to show that ${\pi _0}({L_2}) = {\pi _1}({L_2}) = 0$ where ${L_n}$ is a space of p.l. homeomorphisms of an $n$-simplex, a space introduced by ${\mathbf {R}}$. Thom in his study of the smoothings of combinatorial manifolds.