Common
Universal coefficient theorem
Establish relationships between homology and cohomology theories
Universal coefficient theorem
In algebraic topology, universal coefficient theorems (UCT) establish relationships between homology groups (or cohomology groups) with different coefficients. For instance, for every topological space X, its integral homology groups: H i ( X , Z ) {\displaystyle H_{i}(X,\mathbb {Z} )} completely determine its homology groups with coefficients in A, for any abelian group A: H i ( X , A ) {\displaystyle H_{i}(X,A)} Here H i {\displaystyle H_{i}} might be the simplicial homology, or more generally the singular homology. The usual proof of this result is a pure piece of homological algebra about chain...
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