Cloverleaf representations of simply connected $3$-manifolds

Abstract:

Let $M$ be a triangulated $3$-manifold satisfying the hypothesis of the Poincaré Conjecture. In the present paper it is shown that there is a finite linear graph ${K_1}$ in the $3$-sphere, with exactly two components, and a finite linear graph ${K_2}$ in $M$, such that when the components of the graphs ${K_i}$ are regarded as points, the resulting hyperspaces are homeomorphic. ${K_2}$ satisfies certain conditions which imply that each component of ${K_2}$ is contractible in $M$. Thus the conclusion of the theorem proved here is equivalent to the hypothesis of the Poincaré Conjecture.