Does the Poincaré conjecture refer to π₁ or other fundamental groups?

[pivoxa15]

Does the Poincare conjecture say:

Consider a compact 3-dimensional manifold V without boundary.

Poincare conjectured that
The fundamental group of V is trivial => V is homeomorphic to the 3-dimensional sphere?

It has been proved for all manifolds except 3. However Perelman completed a proof that is almost certainly right for 3-manifolds, thereby proving Poincare to be right.

My question is which fundamental group(s) does the statement refer to? i.e denoting pi for the fundamental group, pi_1 consists of 1 dimensional loops. pi_2 consists of 2-D strips...

 

[mathwonk]

fundamental group always means pi1 not pi2.

 

[pivoxa15]

right. Have I stated the conjecture correctly?