Friday, October 1, 2021

What is the Poincare Conjecture?

Poincaré conjecture is a theorem about the characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dimensional space.

The conjecture states:

“Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere.”

Originally conjectured by Henri Poincaré, the theorem concerns a space that locally looks like ordinary three-dimensional space but is connected, finite in size, and lacks any boundary (a closed 3-manifold).

The Poincaré conjecture claims that if such a space has the additional property that each loop in the space can be continuously tightened to a point, then it is necessarily a three-dimensional sphere.

The analogous conjectures for all higher dimensions were proved before a proof of the original conjecture was found.

Poincaré claimed in 1900 that homology, a tool he had devised based on prior work by Enrico Betti, was sufficient to tell if a 3-manifold was a 3-sphere. However, in a 1904 paper he described a counterexample to this claim, a space now called the Poincaré homology sphere.

The Poincaré sphere was the first example of a homology sphere, a manifold that had the same homology as a sphere, of which many others have since been constructed.

To establish that the Poincaré sphere was different from the 3-sphere, Poincaré introduced a new topological invariant, the fundamental group, and showed that the Poincaré sphere had a fundamental group of order 120, while the 3-sphere had a trivial fundamental group. In this way he was able to conclude that these two spaces were, indeed, different.

This problem seemed to lie dormant until J. H. C. Whitehead revived interest in the conjecture, when in the 1930s he first claimed a proof and then retracted it.

In the process, he discovered some examples of simply-connected (indeed contractible, i.e. homotopically equivalent to a point) non-compact 3-manifolds not homeomorphic to R ^3, the prototype of which is now called the Whitehead manifold.

After nearly a century of effort by mathematicians, Grigori Perelman presented a proof of the conjecture in three papers made available in 2002 and 2003 on arXiv. The proof built upon the program of Richard S. Hamilton to use the Ricci flow to attempt to solve the problem.

Hamilton later introduced a modification of the standard Ricci flow, called Ricci flow with surgery to systematically excise singular regions as they develop, in a controlled way, but was unable to prove this method “converged” in three dimensions. Perelman completed this portion of the proof. Several teams of mathematicians verified that Perelman’s proof was correct.

The Poincaré conjecture, before being proved, was one of the most important open questions in topology.

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