Unknotting cones in the topological category

Ralph Richard Summerhill

Vol. 40 (1972), No. 1, 221–228

DOI: 10.2140/pjm.1972.40.221

Abstract

Let Q be a topological q-manifold, let X be a compact metric space, and let bQ and aX denote the cones over Q and X, respectively. A proper embedding f : aX → bQ (i.e., f(a) = b and f⁻¹[Q] = X) is unknotted if there is homeomorphism h : bQ → bQ such that hf = f, where f is the conical extension of f. In this paper it is proved that a proper embedding is unknotted if and only if bQ − f[aX] and bQ − f⁻[ax] are of the same homotopy type and the embedding f satisfies a local flatness condition.

Mathematical Subject Classification

Primary: 57A35

Milestones

Received: 13 August 1970

Revised: 24 November 1970

Published: 1 January 1972