Embeddings and compressions of polyhedra and smooth manifolds

Abstract

Topology Vol. 4 pp. 361-369. EMBEDDINGS Pergamon Press, 1966. Printed in Great Britain AND COMPRESSIONS OF POLYHEDRA SMOOTH MANIFOLDS? AND MORRIS W. HIRSZH (Receiued 23 July 1965) $1. INTRODUCTION IT IS OFTEN desirable to compress a subset X of a manifold V into a submanifold Y’ by an isotopy of V. For example, V might be a Euclidean space R” and V’ might be p, with k much smaller than n. If X has some special properties ensuring that such a compression is always possible, then we conclude that X embeds in ti. In this article the problem of compressing X into the boundary of an s-cell is studied, where s = dim V. As applications several theorems are proved about embedding polyhedra and smooth manifolds in Eucli- dean space. The main theorems say that a compact polyhedron or smooth submanifold XC V compresses into the boundary of an s-cell provided (1) there exists a “Dehn cone” on X, and (2) V-X is highly connected. A Dehn cone on XC V is an embedding X x I c V with X x 0 = X together with a null homotopy of X x 1 in Y - X x 0. If V-X is sufhci- ently connected, a regular neighborhood theorem of Hudson and Zeeman, together with an engulfing theorem due to Zeeman and the author, provides an s-cell E c V with X c i?E. If V = R’, this implies the compressibility of X into R* -I or Ss -‘. First the piecewise linear (=PL) theory is developed, then the smooth case is reduced to the PL case. In the applications we take X c R4+’ and try to compress X into R4. There are three steps : (1) extend X to X x I c RQ+’ ; (2) choose the extension so that X x 1 3: 0 in RQ+’ - X; (3) prove P+’ - X is sufficiently connected for the Enguhing Theorem to apply. Step (1) is difficult in the PL case, so in the applications it is assumed as part of the hypothesis. In the smooth case, however, (1) is equivalent to the existence of a normal vector field on the smooth submanifold X. Step (2) is accomplished through algebraic topology, sometimes by luck-as when X is a smooth homology sphere-more usually by just assuming that the obstructions to a t This work was supported by the National Science Foundation grant GP-4035