Cohomology is a fundamental concept in algebraic topology and other fields of mathematics that studies the properties of spaces through algebraic invariants. It provides a way to associate a sequence of abelian groups or vector spaces to a topological space, which can help in understanding its structure and features.
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A cohomology theory assigns abelian groups to a topological space . For singular cohomology with coefficients in a commutative ring , form the cochain complex with coboundary and takeA 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 induces cochain pullback and hence : the direction reverses.
The cup product combines degree- and degree- classes into degree , 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 , which has the same additive cohomology groups but zero products between its positive-degree generators.