universal coefficient theorem

{{short description|Establish relationships between homology and cohomology theories}}

In algebraic topology, universal coefficient theorems establish relationships between homology groups (or cohomology groups) with different coefficients. For instance, for every topological space {{mvar|X}}, its integral homology groups:

: $H_i(X,\Z)$

completely determine its homology groups with coefficients in {{mvar|A}}, for any abelian group {{mvar|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 {{mvar|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

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.{{Harv|Kainen|1971}}

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

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 [http://pi.math.cornell.edu/~hatcher/AT/ATpage.html author's homepage].
{{cite journal

| last = Kainen

| first = P. C.

| authorlink = Paul Chester Kainen

| title = Weak Adjoint Functors

| journal = Mathematische Zeitschrift

| volume = 122

| issue =

| pages = 1–9

| year = 1971

| pmid =

| pmc =

| 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

[https://math.stackexchange.com/q/768481 Universal coefficient theorem with ring coefficients]

Category:Homological algebra

Category:Theorems in algebraic topology