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Poincaré conjecture

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.

In the same paper, Poincaré wondered whether a 3-manifold with the homology of a 3-sphere and also trivial fundamental group had to be a 3-sphere. Poincaré's new condition—i.e., "trivial fundamental group"—can be re-phrased as "every loop can be shrunk to a point."

The original phrasing was as follows:

Poincaré never declared whether he believed this additional condition would characterize the 3-sphere, but nonetheless, the statement that it does is known as the Poincaré conjecture . Here is the standard form of the conjecture:

Attempted solutions
This problem seems to have lain dormant for a time, 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 interesting examples of simply connected non-compact 3-manifolds not homeomorphic to R 3, the prototype of which is now called the [Whitehead manifold] .

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