counterexamples to the Poincare conjecture

Nov 2018
South Korea


Sep 2016
I'm not sure what you mean by a simply connected closed 3 manifold being homeomorphic to the Poincare conjecture. How can a manifold be homeomorphic to a conjecture? The conjecture (now a theorem) says that in fact all such manifolds are topological spheres. This means there can't be any counterexamples.

Regarding your examples: Your first example is not a counterexample since it is actually a 2-manifold, not a 3-manifold.

For your second example I'm a bit confused. You claim that "it contains holes but is still simply connected". This is of course impossible. To be more precise, what is typically meant by a "hole" is that there is a nontrivial generator of homology at the 1-level. This could also be interpreted as a nontrivial fundamental homotopy group (in fact this condition is strictly stronger). In any case, neither of these can be true for a simply connected manifold. In fact, the second condition is generally taken as the definition of what it means to be simply connected.