On small maps of manifolds

Hans Samelson

Vol. 15 (1965), No. 4, 1401–1403

DOI: 10.2140/pjm.1965.15.1401

Abstract

A result announced by R. F. Brown in 1963, and completed by Brown and Fadell, generalizing classical results of H. Hopf for differentiable manifolds, is the following:

Theorem: Let M be a compact connected topological manifold; then

(a) M admits arbitrarily small maps with a single fixed point;

(b) If the Euler characteristic χ_M of M is zero, then M admits arbitrarily small maps without fixed points (and conversely). Here a map is small if it is close to the identity map. We propose to give a short proof of this theorem.

Mathematical Subject Classification

Primary: 55.60

Milestones

Received: 8 September 1964

Published: 1 December 1965