Poincaré Conjecture/Dimension 1

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Theorem

Let $\Sigma^1$ be a smooth $1$-manifold.

Let $\Sigma^1$ satisfy:

$H_0 \struct {\Sigma; \Z} = 0$

and:

$H_m \struct {\Sigma; \Z} = \Z$

Then $\Sigma^1$ is homeomorphic to the $1$-sphere $\Bbb S^1$.

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Proof

Follows from the Classification of Compact One-Manifolds.

$\blacksquare$