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 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 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.
0\toHi(X;Z) ⊗ A\overset{\mu}\toHi(X;A)\to\operatorname{Tor}1(Hi-1(X;Z),A)\to0.
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.
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
| 1(H | |
| \operatorname{Ext} | |
| i-1 |
(X;R),G)\toHi(X;G)\overset{h}\to\operatorname{Hom}R(Hi(X;R),G)\to0.
As in the homology case, the sequence splits, though not naturally.
In fact, suppose
Hi(X;G)=\ker\partiali ⊗ G/\operatorname{im}\partiali+1 ⊗ G
and define:
H*(X;G)=\ker(\operatorname{Hom}(\partial,G))/\operatorname{im}(\operatorname{Hom}(\partial,G)).
Then 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 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.
Let, the real projective space. We compute the singular cohomology of with coefficients in using the integral homology, i.e. .
Knowing that the integer homology is given by:
Hi(X;Z)= \begin{cases} Z&i=0ori=nodd,\\ Z/2Z&0<i<n, i odd,\\ 0&otherwise. \end{cases}
We have, so that the above exact sequences yield
\foralli=0,\ldots,n: Hi(X;G)=G.
In fact the total cohomology ring structure is
H*(X;G)=G[w]/\left\langlewn+1\right\rangle.
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.
Hi(X;Z)\cong
| \betai(X) | |
| Z |
⊕ Ti,
where are the Betti numbers of and
Ti
Hi
\operatorname{Hom}(Hi(X),Z)\cong
| \betai(X) | |
| \operatorname{Hom}(Z |
,Z) ⊕ \operatorname{Hom}(Ti,Z)\cong
| \betai(X) | |
| Z |
,
and
\operatorname{Ext}(Hi(X),Z)\cong
| \betai(X) | |
| \operatorname{Ext}(Z |
,Z) ⊕ \operatorname{Ext}(Ti,Z)\congTi.
This gives the following statement for integral cohomology:
Hi(X;Z)\cong
| \betai(X) | |
| Z |
⊕ Ti-1.
For an orientable, closed, and connected -manifold, this corollary coupled with Poincaré duality gives that .
There is a generalization of the universal coefficient theorem for (co)homology with twisted coefficients.
For cohomology we have
| p,q | |
| E | |
| 2=Ext |
| q(H | |
| p(C |
*),G) ⇒ Hp+q(C*;G)
Where
R
C*
R
G
(R,S)
S
Ext
dr
(1-r,r)
Similarly for homology
| 2=Tor | |
| E | |
| p,q |
| R | |
| q(H |
p(C*),G) ⇒ H*(C*;G)
for Tor the Tor group and the differential
dr
(r-1,-r)