Could the Poincaré Conjecture Be False?

Two conjectures are stated which imply that the Poincaré hypothesis (asserting that any simply connected closed compact $3$-manifold is the $3$-sphere) is false. The first one claims that, for certain classes of finitely presented groups, the triviality problem is algorithmically undecidable, and the second one claims that certain embeddings of two-dimensional polyhedra in $3$-manifolds can effectively be constructed.