Cloverleaf representations of simply connected 3-manifolds
Edwin E. Moïse · Transactions of the American Mathematical Society · 1975
Let M M be a triangulated 3 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 {K_1} in the 3 3 -sphere, with exactly two components, and a finite linear graph K 2 {K_2} in M M , such that when the components of the graphs K i {K_i} are regarded as points, the resulting hyperspaces are homeomorphic. K 2 {K_2} satisfies certain conditions which imply that each component of K 2 {K_2} is contractible in M M . Thus the conclusion of the theorem proved here is equivalent to the hypothesis of the Poincaré Conjecture.