Universal coefficient theorem

Consider the tensor product of modules H_i(X; Z) ⊗ A. The theorem states there is a short exact sequence involving the Tor functor

${\displaystyle 0\to H_{i}(X;\mathbf {Z} )\otimes A\,{\overset {\mu }{\to }}\,H_{i}(X;A)\to \operatorname {Tor} _{1}(H_{i-1}(X;\mathbf {Z} ),A)\to 0.}$

Furthermore, this sequence splits, though not naturally. Here μ is a map induced by the bilinear map H_i(X; Z) × A → H_i(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

${\displaystyle 0\to \operatorname {Ext} _{R}^{1}(H_{i-1}(X;R),G)\to H^{i}(X;G)\,{\overset {h}{\to }}\,\operatorname {Hom} _{R}(H_{i}(X;R),G)\to 0.}$

As in the homology case, the sequence splits, though not naturally.

In fact, suppose

${\displaystyle H_{i}(X;G)=\ker \partial _{i}\otimes G/\operatorname {im} \partial _{i+1}\otimes G}$

and define:

${\displaystyle H^{*}(X;G)=\ker(\operatorname {Hom} (\partial ,G))/\operatorname {im} (\operatorname {Hom} (\partial ,G)).}$

Then h above is the canonical map:

${\displaystyle h([f])([x])=f(x).}$

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 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.[1]

Let X = Pⁿ(R), the real projective space. We compute the singular cohomology of X with coefficients in R = Z/2Z.

Knowing that the integer homology is given by:

${\displaystyle H_{i}(X;\mathbf {Z} )={\begin{cases}\mathbf {Z} &i=0{\text{ or }}i=n{\text{ odd,}}\\\mathbf {Z} /2\mathbf {Z} &0<i<n,\ i\ {\text{odd,}}\\0&{\text{otherwise.}}\end{cases}}}$

We have Ext(R, R) = R, Ext(Z, R) = 0, so that the above exact sequences yield

${\displaystyle \forall i=0,\ldots ,n:\qquad \ H^{i}(X;R)=R.}$

In fact the total cohomology ring structure is

${\displaystyle H^{*}(X;R)=R[w]/\left\langle w^{n+1}\right\rangle .}$

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.

${\displaystyle H_{i}(X;\mathbf {Z} )\cong \mathbf {Z} ^{\beta _{i}(X)}\oplus T_{i},}$

where β_i(X) are the Betti numbers of X and ${\displaystyle T_{i}}$ is the torsion part of ${\displaystyle H_{i}}$. One may check that

${\displaystyle \operatorname {Hom} (H_{i}(X),\mathbf {Z} )\cong \operatorname {Hom} (\mathbf {Z} ^{\beta _{i}(X)},\mathbf {Z} )\oplus \operatorname {Hom} (T_{i},\mathbf {Z} )\cong \mathbf {Z} ^{\beta _{i}(X)},}$

and

${\displaystyle \operatorname {Ext} (H_{i}(X),\mathbf {Z} )\cong \operatorname {Ext} (\mathbf {Z} ^{\beta _{i}(X)},\mathbf {Z} )\oplus \operatorname {Ext} (T_{i},\mathbf {Z} )\cong T_{i}.}$

This gives the following statement for integral cohomology:

${\displaystyle H^{i}(X;\mathbf {Z} )\cong \mathbf {Z} ^{\beta _{i}(X)}\oplus T_{i-1}.}$

For X an orientable, closed, and connected n-manifold, this corollary coupled with Poincaré duality gives that β_i(X) = β_n−i(X).

• Allen Hatcher, Algebraic Topology, Cambridge University Press, Cambridge, 2002. ISBN 0-521-79540-0. A modern, geometrically flavored introduction to algebraic topology. The book is available free in PDF and PostScript formats on the author's homepage.
• Kainen, P. C. (1971). "Weak Adjoint Functors". Mathematische Zeitschrift. 122: 1–9. doi:10.1007/bf01113560. S2CID 122894881.

• Universal coefficient theorem with ring coefficients