Vol. 15, No. 4, 1965

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

Jack Max Robertson

Vol. 15 (1965), No. 4, 1383–1389

Let Dn = {x En : |x|1}, and Sn = {x En+1 : |x| = 1}. We denote by Hn the space of C homeomorphisms of Dn onto itself leaving a neighborhood of the boundary fixed. Let Kn 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 Ck topology is used.

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

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

Theorem 2. Kn is arcwise connected for any n.

Mathematical Subject Classification
Primary: 54.80
Received: 16 September 1964
Published: 1 December 1965
Jack Max Robertson