An infinite class of irreducible homology $3$-spheres

Abstract

A class of irreducible homology 3-spheres is obtained by pasting together complements of torus knots. Representations of the fundamental groups of these homology 3-spheres into sym- metric groups are then used to distinguish the members of an in- finite subclass. In this paper, a knot complement means the complement in a 3-sphere of the interior of a regular neighborhood of an imbedded 1-sphere. The objects of concern are certain irreducible homology 3-spheres, each of which is obtained from two disjoint knot comple- ments X and Y by cross-matching meridians and longitudes. That is, one pastes bd(X) to bd(Y) with an orientation reversing homeomor- phism so that a meridian of bd(X) is matched to a longitude of bd(Y) and a longitude of bd(X) is matched to a meridian of bd(Y). W. Browder has suggested to the author that infinitely many of these homology 3-spheres might be topologically distinct. The purpose of this paper is to verify Browder's suggestion. 1. Preliminaries. Let T be a solid torus (of genus one) imbedded in a 3-sphere S. A simple loop in bd(T) which bounds a disk in T but not in bd(T) is called a meridian. A simple loop in bd(T) which does not link the core of T and which does not bound a disk in bd(T) is called a longitude. Equivalently, a longitude is a loop on T which is the boundary of a compact surface in cl(S- T) and which does not bound a disk in bd(T). Evidently, one may discuss meridians and longitudes not only in 3-spheres but also in knot complements. A 3-manifold M is called irreducible if every imbedded 2-sphere in M bounds a 3-cell. LEMMA 1. Let M be the closed 3-manifold which is obtained by cross- matching the meridians and longitudes of two disjoint knot complements X and Y. Then M is an irreducible homology 3-sphere.