Poincaré Conjecture/Dimension 1

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 $\Bbb S^1$.

Proof
Follows from the Classification of Compact One-Manifolds.