Poincaré Conjecture/Dimension 3
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Theorem
Let $\Sigma^3$ be a smooth $3$-manifold.
Let $\Sigma^3$ satisfy:
- $H_0 \struct {\Sigma; \Z} = 0$
and:
- $H_3 \struct {\Sigma; \Z} = \Z$
Then $\Sigma^3$ is homeomorphic to the $3$-sphere $\Bbb S^3$.
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Proof
Follows from Thurston's Geometrization Conjecture.
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$\blacksquare$
Historical Note
The Poincaré Conjecture for a long time was settled for all dimensions but $3$.
Grigori Perelman finally provided the missing piece by demonstrating the truth of Thurston's Geometrization Conjecture in $2003$.