Universal coefficient theorem

Consider the tensor product of modules H_i(X,\Z)\otimes A. The theorem states there is a short exact sequence involving the Tor functor : 0 \to H_i(X, \Z)\otimes A \, \overset{\mu}\to \, H_i(X,A) \to \operatorname{Tor}_1(H_{i-1}(X, \Z),A)\to 0. Furthermore, this sequence splits, though not naturally. Here \mu is the map induced by the bilinear map H_i(X,\Z)\times A\to H_i(X,A). If the coefficient ring A is \Z/p\Z, this is a special case of the Bockstein spectral sequence. ==Universal coefficient theorem for cohomology==

Let G be a module over a principal ideal domain R (for example \Z, or any field.) There is a universal coefficient theorem for cohomology involving the Ext functor, which asserts that there is a natural short exact sequence : 0 \to \operatorname{Ext}_R^1(H_{i-1}(X; R), G) \to H^i(X; G) \, \overset{h} \to \, \operatorname{Hom}_R(H_i(X; R), G)\to 0. As in the homology case, the sequence splits, though not naturally. In fact, suppose :H_i(X;G) = \ker \partial_i \otimes G / \operatorname{im}\partial_{i+1} \otimes G, and define :H^*(X; G) = \ker(\operatorname{Hom}(\partial, G)) / \operatorname{im}(\operatorname{Hom}(\partial, G)). Then h above is the canonical map: :h([f])([x]) = f(x). 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 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. == Example: mod 2 cohomology of the real projective space==

Let X=\mathbb{RP}^n, the real projective space. We compute the singular cohomology of X with coefficients in G=\Z/2\Z using integral homology, i.e., R=\Z. Knowing that the integer homology is given by: :H_i(X; \Z) = \begin{cases} \Z & i = 0 \text{ or } i = n \text{ odd,}\\ \Z/2\Z & 0 We have \operatorname{Ext}(G,G)=G and \operatorname{Ext}(R,G)=0, so that the above exact sequences yield :H^i (X; G) = G for all i=0,\dots,n. In fact the total cohomology ring structure is :H^*(X; G) = G [w] / \left \langle w^{n+1} \right \rangle. ==Corollaries==

A special case of the theorem is computing integral cohomology. For a finite CW complex X, H_i(X,\Z) is finitely generated, and so we have the following decomposition. : H_i(X; \Z) \cong \Z^{\beta_i(X)}\oplus T_{i}, where \beta_i(X) are the Betti numbers of X and T_i is the torsion part of H_i. One may check that : \operatorname{Hom}(H_i(X),\Z) \cong \operatorname{Hom}(\Z^{\beta_i(X)},\Z) \oplus \operatorname{Hom}(T_i, \Z) \cong \Z^{\beta_i(X)}, and :\operatorname{Ext}(H_i(X),\Z) \cong \operatorname{Ext}(\Z^{\beta_i(X)},\Z) \oplus \operatorname{Ext}(T_i, \Z) \cong T_i. This gives the following statement for integral cohomology: : H^i(X;\Z) \cong \Z^{\beta_i(X)} \oplus T_{i-1}. For X an orientable, closed, and connected n-manifold, this corollary coupled with Poincaré duality gives that \beta_i(X)=\beta_{n-i}(X). == Universal coefficient spectral sequence ==

There is a generalization of the universal coefficient theorem for (co)homology with twisted coefficients. For cohomology we have :E^{p,q}_2=\operatorname{Ext}_{R}^q(H_p(C_*),G)\Rightarrow H^{p+q}(C_*;G), where R is a ring with unit, C_* is a chain complex of free modules over R, G is any (R,S)-bimodule for some ring with a unit S, and \operatorname{Ext} is the Ext group. The differential d^r has degree (1-r,r). Similarly for homology, :E_{p,q}^2=\operatorname{Tor}^{R}_q(H_p(C_*),G)\Rightarrow H_*(C_*;G), for \operatorname{Tor} the Tor group and the differential d_r having degree (r-1,-r). == Notes ==