Some topological properties of certain spaces of differentiable homeomorphisms of disks and spheres

Jack Max Robertson

Vol. 15 (1965), No. 4, 1383–1389

DOI: 10.2140/pjm.1965.15.1383

Abstract

Let D_n = {x ∈ E_n : |x|≦ 1}, and Sⁿ = {x ∈ E_n+1 : |x| = 1}. We denote by H_n the space of C^∞ homeomorphisms of D_n onto itself leaving a neighborhood of the boundary fixed. Let K_n be the space of C^∞ orientation preserving homeomorphisms of Sⁿ onto itself. It is not required that maps in the two spaces have differentiable inverses. In both space the C^k topology is used.

The purpose of this paper is to establish the following two theorems:

Theorem 1. H_n is contractible to a point for any n.

Theorem 2. K_n is arcwise connected for any n.

Mathematical Subject Classification

Primary: 54.80

Milestones

Received: 16 September 1964

Published: 1 December 1965