= Nonseparating sphere gives a sphere-circle summand
{title2=$M\cong N\mathbin\#(S^2\times S^1)$}
In an orientable <three-manifold>, an embedded nonseparating two-sphere has a product neighborhood. Join its two boundary spheres by an arc outside that neighborhood and take a regular neighborhood of the union. This region is $S^2\times S^1$ with an open ball removed. Its boundary sphere exhibits the displayed <connected sum>. Consequently a prime but nonirreducible orientable <three-manifold> is $S^2\times S^1$.
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