= Mod-two handle duality
{title2=$H_k(W;\mathbb F_2)\cong H^{n-k}(W,\partial W;\mathbb F_2)$}
Reverse a <handle decomposition> of a compact $n$-dimensional <manifold>. Absolute $k$-handles become relative $(n-k)$-handles based on the boundary. The absolute boundary matrices count <attaching sphere>-<belt sphere> intersections; the corresponding relative matrices are their transposes over $\mathbb F_2$. The absolute <chain complex> is thus the complementary-degree dual of the relative <chain complex>. The <universal coefficient theorem for cohomology> proves <Poincare-Lefschetz duality> without an <orientation> hypothesis.
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