= Poincare duality pairing
{c}
{title2=$\lambda_M(a,b)=\langle a\smile b,[M]\rangle$}
For a closed oriented $d$-dimensional <manifold> over a <field> $F$, the <cup product> pairing
$$
H^q(M;F)\times H^{d-q}(M;F)\to F,\qquad (a,b)\mapsto\langle a\smile b,[M]_F\rangle
$$
is a <perfect pairing> between complementary degrees. The <cap product> with the <fundamental class> is an <isomorphism> by <Poincare duality>, and the <universal coefficient theorem for cohomology> over $F$ identifies the complementary <cohomology> with the full <dual space> of the resulting <homology>. Evaluation is therefore a perfect pairing. Taking the top-degree part of the same formula defines a nondegenerate <bilinear form> on the full graded <cohomology>. A boundary requires relative groups and <Poincare-Lefschetz duality> instead.
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