Poincare Conjecture: Fundamental Group of V Explained

In summary, the Poincare conjecture states that a compact 3-dimensional manifold without boundary is homeomorphic to the 3-dimensional sphere if and only if its fundamental group is trivial. This has been proven for all manifolds except 3, but Perelman's proof for 3-manifolds is believed to be correct. The statement refers to the fundamental group pi1, which consists of 1-dimensional loops. Pi2, which consists of 2-dimensional strips, is not relevant in this conjecture.
  • #1
pivoxa15
2,255
1
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...
 
Last edited:
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  • #2
fundamental group always means pi1 not pi2.
 
  • #3
right. Have I stated the conjecture correctly?
 

What is the Poincare Conjecture?

The Poincare Conjecture is a mathematical problem that was first proposed by French mathematician Henri Poincare in the early 1900s. It states that any closed, simply connected 3-dimensional manifold is topologically equivalent to a 3-sphere.

What is the Fundamental Group of V?

The Fundamental Group of V, also known as the fundamental group of a space, is a mathematical concept used to describe the structure of a space. It is a group that consists of all possible loops within the space, where the operation is loop concatenation.

Why is the Poincare Conjecture considered a fundamental problem in mathematics?

The Poincare Conjecture is considered a fundamental problem in mathematics because it is a fundamental question about the topology of 3-dimensional spaces. It has implications in many areas of mathematics, including geometry, topology, and differential equations.

What progress has been made in solving the Poincare Conjecture?

In 2002, Russian mathematician Grigori Perelman proposed a proof for the Poincare Conjecture, building upon the work of several other mathematicians. In 2006, he was awarded the Fields Medal and the Clay Mathematics Institute Millennium Prize for his contribution. However, his proof has not yet been fully accepted by the mathematical community.

What are the potential applications of solving the Poincare Conjecture?

Solving the Poincare Conjecture could have significant implications in the fields of physics and engineering, particularly in the study of 3-dimensional spaces and their properties. It could also lead to a better understanding of the fundamental nature of our universe.

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