= Coprime two-cell attachments to a circle
{title2=$S^1\cup_{m}D^2\cup_{n}D^2\simeq S^2\quad(\gcd(m,n)=1)$}
The cellular boundary $\mathbb Z^2\to\mathbb Z$ is $(m,n)$, with kernel $\mathbb Z$ and zero cokernel when the degrees are coprime. The <van Kampen theorem> gives trivial fundamental group. The <Hurewicz theorem> supplies a map from the sphere representing a homology generator, and the <homological Whitehead theorem> makes it a homotopy equivalence. For degrees $2,3$ the space cannot be homeomorphic to a sphere: an embedded circle in a sphere bounds its two complementary regions, so both characteristic boundary maps would have degree of absolute value one.
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