Universal coefficient theorem

In algebraic topology, universal coefficient theorems establish relationships between homology and cohomology theories. For instance, the integral homology theory of a topological space $X$ , and its homology with coefficients in any abelian group $A$ are related as follows: the integral homology groups

H i (X; Z)

completely determine the groups

H i (X; A)

Here $H i$ might be the simplicial homology or more general singular homology theory: 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 $A$ 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

0 \to H_i(X; \mathbf{Z})\otimes A \overset{\mu}\to H_i(X;A) \to \mbox{Tor}(H_{i-1}(X; \mathbf{Z}),A)\to 0.

Furthermore, this sequence splits, though not naturally. Here $μ$ is a map induced by the bilinear map $\times$ .

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 $R$ (e.g., $Z$ 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{Ext}_R^1(\operatorname{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 from $X$ 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. ^[1]

Example: mod 2 cohomology of the real projective space

Let $X = P n (R)$ , the real projective space. We compute the singular cohomology of $X$ with coefficients in $R = Z /2 Z$ .

Knowing that the integer homology is given by:

H_i(X; \mathbf{Z}) = \begin{cases} \mathbf{Z} & i = 0 \mbox{ or } i = n \mbox{ odd,}\\ \mathbf{Z}/2\mathbf{Z} & 0<i<n,\ i\ \mbox{odd,}\\ 0 & \mbox{else.} \end{cases}

We have $Ext(R, R) = R, Ext(Z, R) = 0$ , so that the above exact sequences yield

\forall i = 0, \cdots, n: \qquad \ H^i (X; R) = R.

In fact the total cohomology ring structure is

H^*(X; R) = R [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; \mathbf{Z}) \cong \mathbf{Z}^{\beta_i(X)}\oplus T_{i},

where $β i (X)$ are the betti numbers of $X$ and $T_i$ is the torsion part of $H_i$ . One may check that

\mbox{Hom}(H_i(X),\mathbf{Z}) \cong \mbox{Hom}(\mathbf{Z}^{\beta_i(X)},\mathbf{Z}) \oplus \mbox{Hom}(T_i, \mathbf{Z}) \cong \mathbf{Z}^{\beta_i(X)},

and

\mbox{Ext}(H_i(X),\mathbf{Z}) \cong \mbox{Ext}(\mathbf{Z}^{\beta_i(X)},\mathbf{Z}) \oplus \mbox{Ext}(T_i, \mathbf{Z}) \cong T_i.

This gives the following statement for integral cohomology:

H^i(X;\mathbf{Z}) \cong \mathbf{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 $β i (X) = β n - i (X)$ .

Notes

↑ (Kainen 1971)

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 author's homepage.

Kainen, P. C. (1971). "Weak Adjoint Functors". Mathematische Zeitschrift 122: 1–9. doi:10.1007/bf01113560.

External links

http://math.stackexchange.com/questions/767864/universal-coefficient-theorem-with-ring-coefficients/768481#768481

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