Handle-theory and engulfing

Abstract

This paper is concerned with the engulfing of a polyhedron X from one end δ_ W of a cobordism (W, δ_ W, δ+W). In the case where the pair (W, δ_ W) is highly connected this version of Engulfing is dealt with in Rourke and Sanderson (2) by combining the method used in the proof of the h-cobordism theorem (eliminating handles) with a simple procedure involving handle-moves. A handle-move is an ambient isotopy which shrinks a handle H onto a small regular neighbourhood of its fibre D. Thus, if a closed polyhedron misses D, a handle-move can be applied to cause X to slip off H.