Universal coefficient theorem

In algebraic topology, universal coefficient theorems establish relationships between homology groups (or cohomology groups) with different coefficients. For instance, for every topological space X, its integral homology groups: Hi(X; Z) completely determine its homology groups with coefficients in A, for any abelian group A: Hi(X; A) Here Hi 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 complexes of free abelian groups. The form of the result is that other coefficients A may be used, at the cost of using a Tor functor. For example it is common to take A to be Z/2Z, so that coefficients are modulo 2. This becomes straightforward in the absence of 2-torsion in the homology. Quite generally, the result indicates the relationship that holds between the Betti numbers bi of X and the Betti numbers bi,F with coefficients in a field F. These can differ, but only when the characteristic of F is a prime number p for which there is some p-torsion in the homology. Consider the tensor product of modules Hi(X; Z) ⊗ A. The theorem states there is a short exact sequence involving the Tor functor Furthermore, this sequence splits, though not naturally. Here μ is the map induced by the bilinear map Hi(X; Z) × A → Hi(X; A). If the coefficient ring A is Z/pZ, this is a special case of the Bockstein spectral sequence. Let G be a module over a principal ideal domain R (e.g., Z or a field.) There is also a universal coefficient theorem for cohomology involving the Ext functor, which asserts that there is a natural short exact sequence As in the homology case, the sequence splits, though not naturally. In fact, suppose and define: Then h above is the canonical map: An alternative point-of-view can be based on representing cohomology via Eilenberg–MacLane space where the map h takes a homotopy class of maps from X to K(G, i) to the corresponding homomorphism induced in homology. Thus, the Eilenberg–MacLane space is a weak right adjoint to the homology functor.

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