On Realizing Centralizers of Certain Elements in the Fundamental Group of a 3-manifold

The main result in this note is that if X is a simple loop in the boundary of a compact, irreducible, orientable 3-manifold M and [X] ¥= 1 G trx(M), one can represent the centralizer of [X] in irx(M) by a Seifert fibred submanifold of M. Introduction. The main result in this note is a partial answer to a question of Jaco [2]. Jaco has shown in [2] that the centralizer of a nontrivial element in the fundamental group of a sufficiently large compact, orientable 3manifold is isomorphic to the fundamental group of a Seifert fibre space. He also suggests that one might geometrically realize this group by a submanifold of the ambient manifold. It is the purpose of this note to show that the above realization can in fact be made if the element is represented by a simple loop in the boundary of the 3-manifold and the 3-manifold is irreducible. Proposition 7.1 in [2] is quite similar to our Theorem 2. Our notation and definitions are standard unless otherwise indicated. We say that a manifold A is properly embedded in a manifold M if A n dM = 3A. Let A be an annulus. A spanning arc a of A is an arc properly embedded in A such that A — a is simply connected. Throughout the remainder of this paper a will denote a spanning arc of A. Proposition 1. Let Ax, . . . , Am be a collection of annuli properly embedded in M such that Aj n Aj =M, =dAjfor 1 < j < / < m. Let f: (A,dA)^>(M, dM) be a map such that (l)f(dA) = dAx, (2)/^: irx(A) —> trx(M) is monk, and (3)f(a) is not homotopic rel its boundary to an arc in (J Tli-^/Then there is an embedding g: (A, dA)^>(M, dM) such that (\)g(A)n Aj=dAjfori= \,...,m, (2) gt: trx(A) —> ttx(M) is monic, (3) g(a) is not homotopic rel its boundary to an arc in U ", XA,. Proof. We show first that f\ dA may be assumed to be an embedding. Let C, and C2 be the components of dA. Let (M, p) be the covering space of M associated with the subgroup of ttx(M) generated by the class of the simple loop/(C,). Sincef%ttx(A) C p^ttx(M), there is a map/: (A, dA)->(M, dM) such that pf = f. It is a consequence of the theorem in [8] that there is an embedding/,: (A, dA)^(M, dM) such that Jx(dA) = /(dA). We may Received by the editors October 28, 1974 and, in revised form, March 3, 1975. AMS (MOS) subject classifications (1970). Primary 55A05; Secondary 57A35, 57A65, 57C35.