= Cohomology
{title2=$H^*(X;R)$}
A <cohomology> theory assigns <abelian groups> $H^q(X;R)$ to a <topological space> $X$. For <singular cohomology> with coefficients in a commutative <ring> $R$, form the <cochain complex> $C^q(X;R)$ with <coboundary> $\delta$ and take
$$
H^q(X;R)=\ker(\delta:C^q\to C^{q+1})/\operatorname{im}(\delta:C^{q-1}\to C^q).
$$
A <cocycle> represents a <cohomology class>; adding a <coboundary> leaves that class unchanged.
A <cohomology class> evaluates on a <homology class> by applying its <cochain> to a representing <cycle>. Replacing either representative changes the result by zero, explaining the relation to <homology>. A <continuous map> $f:X\to Y$ induces <cochain pullback> and hence $f^*:H^q(Y;R)\to H^q(X;R)$: the direction reverses.
The <cup product> combines degree-$p$ and degree-$q$ classes into degree $p+q$, giving a graded <cohomology ring>. The <circle> has one integral generator in degrees zero and one. A two-dimensional <torus> has two independent degree-one classes whose <cup product> generates degree two. That multiplication distinguishes it from $S^1\vee S^1\vee S^2$, which has the same additive <cohomology groups> but zero products between its positive-degree generators.
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