Research Announcements

1. Let M be a connected, orientable, triangulated 3-manifold and let A(Zw2(M) be a sub-group which is invariant under the operators in 7Ti(il/f). By a Ti(M)-class in 7r2(.M) we mean the set of elements ±£a, for some a£7r2( if) and every %ÇÎTI(M). The invariance under 7Ti(M) means that, if a £ A , then the entire 7Ti(Af)-class {+£#} is contained in À. A wi(M)-class is represented by a map 5—»M", without reference to base-points or orientation. We shall describe such a map, or singular sphere, as essential mod A if, and only if, the corresponding 7Ti(AT)-class is not contained in A. The terms polyhedral, piecewise linear etc., when applied to My will refer to the piecewise affine structure which M derives from the given triangulation K. Thus a polyhedron in M is the carrier of a subcomplex, L, of a (rectilinear) subdivision of K. The polyhedron is compact if, and only if, L is a finite complex. The main purpose of this note is to show how the proof of the (qualified) sphere theorem, due to C. D. Papakyriakopoulos [5], can be modified so as to yield a proof of: