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'':

:

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

:

Here might be the simplicial homology, or more generally the singular homology: 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 may be used, at the cost of using a Tor functor.

For example it is common to take to be , 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 of and the Betti numbers with coefficients in a field . These can differ, but only when the characteristic of is a prime number for which there is some -torsion in the homology.

Statement of the homology case

Consider the tensor product of modules . The theorem states there is a short exact sequence involving the Tor functor

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

Furthermore, this sequence splits, though not naturally. Here is the map induced by the bilinear map .

If the coefficient ring is , this is a special case of the Bockstein spectral sequence.

Universal coefficient theorem for cohomology

Let be a module over a principal ideal domain (e.g., 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

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

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 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 takes a homotopy class of maps from to 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 , the real projective space. We compute the singular cohomology of with coefficients in .

Knowing that the integer homology is given by:

:H_i(X; \mathbf) =
\begin
\mathbf & i = 0 \text i = n \text\\
\mathbf/2\mathbf & 0

We have , 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 / \left \langle w^ \right \rangle.

Corollaries

A special case of the theorem is computing integral cohomology. For a finite CW complex , is finitely generated, and so we have the following decomposition.

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

where are the Betti numbers of and $T_i$ is the torsion part of $H_i$. One may check that

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

and

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

This gives the following statement for integral cohomology:

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

For an orientable, closed, and connected -manifold, this corollary coupled with Poincaré duality gives that .

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

External links

Universal coefficient theorem with ring coefficients

Homological algebra
Theorems in algebraic topology