The Universal Coefficient Theorem establishes the relationship between the homology or cohomology groups of a topological space computed with different coefficient groups, showing that a space's integral homology groups completely determine its homology groups with any other coefficient group, up to a correction term supplied by the Tor functor of homological algebra. The theorem also relates the Betti numbers of a space computed over the rational numbers to those computed over a field of prime characteristic, showing the two can differ only in dimensions where the space's homology carries torsion of that same prime.
Facts
StatementFor every topological space X and abelian group A there is a short exact sequence 0 -> H_i(X,Z) tensor A -> H_i(X,A) -> Tor_1(H_{i-1}(X,Z),A) -> 0, which splits though not naturally, so integral homology completely determines homology with coefficients in A. 1 Classification
Statement Form Connections
Has Statement Form
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
In Branch
Sources
1. Universal coefficient theorem (Wikipedia)
Statement of the homology caseQuote, Statement of the homology case
The theorem states there is a short exact sequence involving the Tor functor
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