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 , its ''integral homology groups'':

:$H_i(X,\Z)$

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

:$H_i(X,A)$

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 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 $b_$ 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

:$0 \to H_i(X, \Z)\otimes A \, \overset\to \, H_i(X,A) \to \operatorname_1(H_(X, \Z),A)\to 0.$

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

:$0 \to \operatorname_R^1(H_(X; R), G) \to H^i(X; G) \, \overset \to \, \operatorname_R(H_i(X; R), G)\to 0.$

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

:$H_i(X;G) = \ker \partial_i \otimes G / \operatorname\partial_ \otimes G,$

and define

:$H^*(X; G) = \ker(\operatorname(\partial, G)) / \operatorname(\operatorname(\partial, G)).$

Then $h$ above is the canonical map:

:h( ( = 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 $X=\mathbb^n$, 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
\Z & i = 0 \text i = n \text\\
\Z/2\Z & 0

We have $\operatorname(G,G)=G$ and $\operatorname(R,G)=0$, 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 / \left \langle w^ \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.

:$H_i(X; \Z) \cong \Z^\oplus T_,$

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

:$\operatorname(H_i(X),\Z) \cong \operatorname(\Z^,\Z) \oplus \operatorname(T_i, \Z) \cong \Z^,$

and

:$\operatorname(H_i(X),\Z) \cong \operatorname(\Z^,\Z) \oplus \operatorname(T_i, \Z) \cong T_i.$

This gives the following statement for integral cohomology:

:$H^i(X;\Z) \cong \Z^ \oplus T_.$

For $X$ an orientable, closed, and connected $n$-manifold, this corollary coupled with Poincaré duality gives that $\beta_i(X)=\beta_(X)$.

Universal coefficient spectral sequence

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

For cohomology we have

:$E^_2=\operatorname_^q(H_p(C_*),G)\Rightarrow H^(C_*;G),$

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

Similarly for homology,
:$E_^2=\operatorname^_q(H_p(C_*),G)\Rightarrow H_*(C_*;G),$
for $\operatorname$ the Tor group and the differential $d_r$ having degree $(r-1,-r)$.

Notes

References

* Allen Hatcher, ''Algebraic Topology'', Cambridge University Press, Cambridge, 2002. . A modern, geometrically flavored introduction to algebraic topology. The book is available free in PDF and PostScript formats on th
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

Universal coefficient theorem with ring coefficients

Homological algebra
Theorems in algebraic topology