= Simply connected closed three-manifolds are homology spheres
{title2=$H_i(M;\mathbb Z)\cong H_i(S^3;\mathbb Z)$}
A simply connected closed three-dimensional <manifold> is orientable, since its orientation monodromy factors through its trivial <fundamental group>. Its first integral homology is the <abelianization> of that fundamental group and is zero. The <universal coefficient theorem for cohomology> gives $H^1(M;\mathbb Z)=0$, and <Poincare duality> gives $H_2=0$, $H_3=\mathbb Z$. Connectedness gives $H_0=\mathbb Z$, and duality gives zero homology above dimension three. Thus it is an integral <homology sphere>, without needing any homeomorphism classification.
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