Universal coefficient theorem

Template:Short description In algebraic topology, universal coefficient theorems establish relationships between homology groups (or cohomology groups) with different coefficients. For instance, for every topological space Template:Mvar, its integral homology groups:

$H_i(X,\mathbb{Z})$

completely determine its homology groups with coefficients in Template:Mvar, for any abelian group Template:Mvar:

$H_i(X,A)$

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 Template:Mvar may be used, at the cost of using a Tor functor.

For example, it is common to take $A$ to be $\mathbb{Z}/2\mathbb{Z}$, 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 $H_i(X,\mathbb{Z})\otimes A$. The theorem states there is a short exact sequence involving the Tor functor

$0\to H_i(X,\mathbb{Z})\otimes A\stackrel{\mu }{\to }{H}_i(X,A)\to {\mathrm{Tor}}_1(H_{i-1}(X,\mathbb{Z}),A)\to 0.$

Furthermore, this sequence splits, though not naturally. Here $\mu$ is the map induced by the bilinear map $H_i(X,\mathbb{Z})\times A\to H_i(X,A)$.

If the coefficient ring $A$ is $\mathbb{Z}/p\mathbb{Z}$, 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 $\mathbb{Z}$, 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

$0\to {\mathrm{Ext}}_R^1(H_{i-1}(X;R),G)\to H^i(X;G)\stackrel{h}{\to }{\mathrm{Hom}}_R(H_i(X;R),G)\to 0.$

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

$H_i(X;G)=\ker {\partial }_i\otimes G/\mathrm{im}{\partial }_{i+1}\otimes G,$

and define

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

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 $X={ℝ\mathbb{P}}^n$, the real projective space. We compute the singular cohomology of $X$ with coefficients in $G=\mathbb{Z}/2\mathbb{Z}$ using integral homology, i.e., $R=\mathbb{Z}$.

Knowing that the integer homology is given by:

$H_i(X;\mathbb{Z})=\{\begin{array}{cc}\mathbb{Z}& i=0\text{ or }i=n\text{ odd,}\\ \mathbb{Z}/2\mathbb{Z}& 0<i<n,i\text{odd,}\\ 0& \text{otherwise.}\end{array}$

We have $\mathrm{Ext}(G,G)=G$ and $\mathrm{Ext}(R,G)=0$, 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

$H^{*}(X;G)=G[w]/⟨w^{n+1}⟩.$

Corollaries

A special case of the theorem is computing integral cohomology. For a finite CW complex $X$, $H_i(X,\mathbb{Z})$ is finitely generated, and so we have the following decomposition.

$H_i(X;\mathbb{Z})\cong {\mathbb{Z}}^{{\beta }_i(X)}\oplus T_i,$

where ${\beta }_i(X)$ are the Betti numbers of $X$ and $T_i$ is the torsion part of $H_i$. One may check that

$\mathrm{Hom}(H_i(X),\mathbb{Z})\cong \mathrm{Hom}({\mathbb{Z}}^{{\beta }_i(X)},\mathbb{Z})\oplus \mathrm{Hom}(T_i,\mathbb{Z})\cong {\mathbb{Z}}^{{\beta }_i(X)},$

and

$\mathrm{Ext}(H_i(X),\mathbb{Z})\cong \mathrm{Ext}({\mathbb{Z}}^{{\beta }_i(X)},\mathbb{Z})\oplus \mathrm{Ext}(T_i,\mathbb{Z})\cong T_i.$

This gives the following statement for integral cohomology:

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

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

$E_2^{p,q}={\mathrm{Ext}}_R^q(H_p(C_{*}),G)\Rightarrow H^{p+q}(C_{*};G),$

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 $\mathrm{Ext}$ is the Ext group. The differential $d^r$ has degree $(1-r,r)$.

Similarly for homology,

$E_{p,q}^2={\mathrm{Tor}}_q^R(H_p(C_{*}),G)\Rightarrow H_{*}(C_{*};G),$

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

References

• Allen Hatcher, Algebraic Topology, Cambridge University Press, Cambridge, 2002. Template:ISBN. A modern, geometrically flavored introduction to algebraic topology. The book is available free in PDF and PostScript formats on the author's homepage.
• Template:Cite journal
• Jerome Levine. “Knot Modules. I.” Transactions of the American Mathematical Society 229 (1977): 1–50. https://doi.org/10.2307/1998498

External links

• Universal coefficient theorem with ring coefficients