= Commutator handlebody of the torus
{title2=$T^2\times D^2$}
Attach two <one-handles> to a four-dimensional <zero-handle>, then a <two-handle> along the commutator of their generators with the product zero framing. The result is $T^2\times D^2$. Its spine is the usual cell structure of the <torus>: one vertex, two one-cells, and a two-cell attached by $xyx^{-1}y^{-1}$. An embedded representative of its $H_2$ generator is obtained by taking a once-punctured torus carried by the two one-handles and capping its boundary by the core disk of the two-handle. It meets the relative co-core once and has self-intersection zero. The boundary is $T^3$, whose integral homology has ranks $1,3,3,1$.
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