Poincare Conjecture: Fundamental Group of V Explained

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SUMMARY

The Poincaré Conjecture asserts that for a compact 3-dimensional manifold V without boundary, if the fundamental group π₁ of V is trivial, then V is homeomorphic to the 3-dimensional sphere. This conjecture has been proven for all manifolds except for 3-manifolds, with Grigori Perelman providing a proof that is widely accepted as correct. The discussion clarifies that the term "fundamental group" typically refers to π₁, which consists of 1-dimensional loops, rather than π₂, which pertains to 2-dimensional surfaces.

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pivoxa15
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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...
 
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fundamental group always means pi1 not pi2.
 
right. Have I stated the conjecture correctly?
 

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