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

Abstract

In his study of the smoothings of p. l. manifolds, R. Thom considered the homotopy groups of a certain space L n {L_n} of p.l. homeomorphisms on an n n -simplex. N. H. Kuiper showed in 1965 that the higher homotopy groups of L n {L_n} were in general nontrivial. The main result in this paper is that π 0 ( L 2 ) = π 1 ( L 2 ) = 0 {\pi _0}({L_2}) = {\pi _1}({L_2}) = 0 . The proof of this result is based on a theorem of S. S. Cairns in 1944 on the deformation of rectilinear complexes in R 2 {R^2} and a theorem established in Part I of this paper.