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:

H i (X; Z)

completely determine its homology groups with coefficients in $A$ , for any abelian group $A$ :

H i (X; A)

Here $H i$ 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 $Z /2 Z$ , 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[edit]

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;\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 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[edit]

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

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 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[edit]

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

Knowing that the integer homology is given by:

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

\forall i=0,\ldots ,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[edit]

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

\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

\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 $β i (X) = β n - i (X)$ .

Universal coefficient spectral sequence[edit]

There is a generalization of the universal coefficient theorem for (co)homology with twisted coefficients.

For cohomology we have

E_{2}^{p,q}=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 $S$ , $Ext$ is the Ext group. The differential $d^{r}$ has degree $(1-r,r)$ .

Similarly for homology

E_{p,q}^{2}=Tor_{q}^{R}(H_{p}(C_{*}),G)\Rightarrow H_{*}(C_{*};G)

for Tor the Tor group and the differential $d_{r}$ having degree $(r-1,-r)$ .

Notes[edit]

^ (Kainen 1971)

References[edit]

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.
Jerome Levine. “Knot Modules. I.” Transactions of the American Mathematical Society 229 (1977): 1–50. https://doi.org/10.2307/1998498

External links[edit]

Universal coefficient theorem with ring coefficients

[1] (Kainen 1971)

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