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

Abstract

Let K K be a proper rectilinear triangulation of a 2 2 -simplex S S in the plane and L ( K ) L(K) be the space of all homeomorphisms of S S which are linear on each simplex of K K and are fixed on Bd ( S ) \text {Bd}(S) . The author shows in this paper that L ( K ) 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 ) L(K) is pathwise connected. Both results will be used in Part II of this paper to show that π 0 ( L 2 ) = π 1 ( L 2 ) = 0 {\pi _0}({L_2}) = {\pi _1}({L_2}) = 0 where L n {L_n} is a space of p.l. homeomorphisms of an n n -simplex, a space introduced by R {\mathbf {R}} . Thom in his study of the smoothings of combinatorial manifolds.