Universal Coefficient Theorem

Algebraic Topology · Axiom Academy

LESSON Universal Coefficient Theorem Connecting Homology and Cohomology through Short Exact Sequences 1. The Universal Coefficient Theorem For any space X and abelian group G, there exists a natural short exact sequence: This sequence splits (but not naturally), giving us the isomorphism: Ext(H_ n-1 (X), G): Measures the "extension" obstruction - how much H^n fails to be just Hom(H_n(X), G) Hom(H_n(X), G): The main term - homomorphisms from homology to G Splitting: The sequence splits, but the splitting is not canonical (not functorial) 2. Understanding the Ext Functor The Ext functor measures how far an abelian group is from being free. For abelian groups A and G: Ext^1(A, G) = 0 if and only if A is free (or more generally, projective) Ext^1(Z/nZ, G) = G/nG - for finite cyclic groups, Ext picks out the n-torsion Ext^1(A ⊕ B, G) = Ext^1(A, G) ⊕ Ext^1(B, G) - Ext is additive The Ext term in the UCT captures torsion phenomena and extension problems between homology and cohomology. 3. When H^n(X; G) ≅ Hom(H_n(X), G) The UCT simplifies dramatically in several important cases where the Ext term vanishes: If H_ n-1 (X) is free abelian, then Ext(H_ n-1 (X), G) = 0, giving: When G is a field F (like Z/2Z or R), we always have: This is because Ext(A, F) = 0 for any abelian group A and field F. For spaces where all homology groups are torsion-free (like spheres S^n, tori T^n, or simply connected CW complexes), the Ext term always vanishes.

This is the written version of the interactive lesson above. See the full Algebraic Topology course.