Some applications of Waldhausen’s results on irreducible surfaces
Charles D. Feustel · Transactions of the American Mathematical Society · 1970
FEUSTEL 0. Preliminaries.In this paper we will use a number of results of Waldhausen in [10] to partially answer three questions of Neuwirth posed in [5].Throughout this paper all spaces will be simplicial complexes and all maps will be piecewise linear.We will denote the boundary, closure, and interior of a set X by bd {X), cl {X), and int {X) respectively.We denote the closed unit interval by /.The author wishes to thank L. S. Husch and N. Max for conversations. 1. Certain maximal subgroups of II1(A/3) and a problem of Neuwirth.In this section we will show that a subgroup of II i(M3) associated with a closed surface embedded in M3 is in some sense maximal.We shall follow Kneser in saying M3 is irreducible if every embedded 2-sphere bounds an embedded 3-ball.Theorem 1.Let Tn and Tm be compact closed connected, orientable surfaces.Let M3 be an orientable 3-manifold which is irreducible.Let f be an embedding of Tn in M3 such that f*: F[A[Tn, y) -*■ Il^M3, x) is a monomorphism.If fl^T,,,) is isomorphic to a subgroup A^U^M3, x) and ^=>/*(n1(7,n, >>)), then Tn is homeomorphic to Tm andf^U^Tn, y) = A. Corollary 1.Let k be a knot.Let X= S3 -k and P be a peripheral subgroup of n1(A').If A is an abelian subgroup ofU^X) containing P, then A =P(1).Proof of corollary.This is a consequence of a theorem of Papakyriakopoulos that all abelian subgroups of a knot group are subgroups of Z © Z (Theorem (5.4.2) on p. 56 of [5]) and the fact that X is irreducible.The proof of the theorem will be divided into four lemmas and a theorem.The desired result is then an easy consequence of 1.4 and 1.5.We will follow Waldhausen in saying that a surface 5 contained in a manifold M is incompressible in M if for each component S( of S the natural map from II1(5'j) into fli{M) is an injection and St is not the 2-sphere.Lemma 1.1.Let M3 be a compact, connected, orientable, irreducible 3-manifold with incompressible boundary.Let Tn be a compact, closed, connected, orientable surface.Let fl^r,,)^ UX{M3).Then M3 is homeomorphic to Tn x /.