Isotopic Unknotting in F × I

Abstract

This paper is essentially a generalization of Unknotting in M1 X /, by E. M. Brown. The major results in this paper concern the existence of ambient isotopies of unknotted arcs (families of arcs) properly embedded in F x /. I. Introduction. This paper is essentially a generalization of (1). Loosely speaking when Brown shows the existence of a homeomorphism having certain properties, we show the existence of an isotopy having many of the same properties. Our major results concern the existence of an ambient isotopy of unknotted arcs in F X I. The results in this paper come from the author's doctoral thesis at Dartmouth College; however the proofs have been simplified by following some of the techniques used by Waldhausen in (6). We also adopt much of the terminology used in (6). The author would like to thank E. M. Brown of Dartmouth College for directing his dissertation and suggesting these prob- lems. The author would also like to thank the referee for a number of helpful suggestions and for pointing out the necessity of an added hypothesis in the statement of Lemma 3.4. II. Notation. In this paper all spaces are simplicial complexes and all maps are piecewise linear. We shall denote the boundary, closure, and interior of a subspace A of a space F by bd(A, F), cl(A, F), and int(A, F) respectively. Since no confusion can result, we will be able to abbreviate these by bd(A), cl(A), and int(A). The symbol F will denote a compact, connected surface and / the unit interval. A surface F is properly embedded in a 3-manifold M if F n bd(M) = bd(F). A surface F properly embedded in a 3-manifold M is incompressible in M if for every disk D embedded in M such that D n F = bd(D), bd(L>) is nullhomotopic in F Let F be a two-sided surface properly embedded in the 3-manifold M. Then the manifold M', obtained by splitting M along F, has by definition the properties: