Universal coefficient theorem

In algebraic topology, universal coefficient theorems establish relationships between homology and cohomology theories. For instance, the integral homology theory of a topological space X, and its homology with coefficients in any abelian group A are related as follows: the integral homology groups

H_i(X; Z)

completely determine the groups

H_i(X; A)

Here H_i might be the simplicial homology or more general singular homology theory: the result itself 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 b_i of X and the Betti numbers b_i,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.

Statement of the homology case

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

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

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.

Universal coefficient theorem for cohomology

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

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

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 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]

Example: mod 2 cohomology of the real projective space

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:

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

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

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

In fact the total cohomology ring structure is

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

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.

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

where β_i(X) are the Betti numbers of X and $T_{i}$ is the torsion part of $H_{i}$. 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:

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

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

Notes

1. (Kainen 1971)

References

• 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.

External links

• http://math.stackexchange.com/questions/767864/universal-coefficient-theorem-with-ring-coefficients/768481#768481