A “piecewise-linear” method for triangulating 3-manifolds

Abstract

In [ 121, E. E. Moise proved a piecewise-linear approximation theorem for homeomorphisms between 3-manifolds, which implies the triangulation theorem and Hauptvermutung for 3-manifolds without boundary. Moise’s proof, and the later proof given by Bing [3], depend on deep point-settheoretic techniques. The present author, in his Ph.D. thesis (Harvard, Fall 1971), gave a proof of this approximation theorem in which all the work was done in the piecewise-linear (PL) category; the main tools were the so-called Dehn lemma and the loop theorem, fundamental piecewise-linear results due to the late C. D. Papakyriakopoulos, which had been unavailable when [ 121 and [3] were written. Since then other proofs have appeared in 14, 7, 14 ]; all use the Dehn lemma and the loop theorem in combination with various point-set-theoretic methods or more advanced PL methods. The author’s original goal in the research leading to his thesis was to provide a proof of the theorem that would seem natural, straightforward and intuitively comprehensible from the viewpoint of standard 3-dimensional PL topology, and would use nothing more difficult than Dehn’s lemma and the loop theorem. He hopes that the present paper, which is a new piece of work incorporating many ideas from his thesis, achieves that goal. The theorem that is proved here is actually a refinement of Moise’s original result, and was first proved by Bing [ 2) and independently by Moise [ 13 ]. If E is a positive-valued continuous function on a space X, and iff and g are maps of X into a space Y(distance function) d, we say that g Eapproximates f (or is an s-approximation to f) if d(f(x), g(x)) < E(X) for every x E X. An embedding is a l-l map (not necessarily “proper”). A mamyold may have a boundary.