= Cycle class map
{title2=$\operatorname{cl}:\operatorname{CH}^r(X)\to H^{2r}(X,\mathbb Z)$}
For a smooth complex projective <algebraic variety>, the cycle class map sends an <algebraic cycle> of codimension $r$ to the <Poincare dual> of its homology class. <Rational equivalence> gives the same cohomology class, so the map descends to the <Chow group>. It respects <intersection products>, pullback and proper pushforward; a cohomological obstruction therefore obstructs the corresponding proposed identity of <Chow classes>.
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