Universal coefficient theorem

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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

Hi(X; Z)

completely determine the groups

Hi(X; A)

Here Hi 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/2Z, 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 bi of X and the Betti numbers bi,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[edit]

Consider the tensor product of modules Hi(X; Z) ⊗ 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 Hi(X; Z) ⊗ AHi(X; A).

If the coefficient ring A is Z/pZ, this is a special case of the Bockstein spectral sequence.

Universal coefficient theorem for cohomology[edit]

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[edit]

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

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[edit]

A special case of the theorem is computing integral cohomology. For a finite CW complex X, Hi(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 βi(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 βi(X) = βni(X).

Notes[edit]

References[edit]

  • 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.

External links[edit]