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 X, its integral homology groups:

Hi(X; Z)

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

Hi(X; A)

Here Hi 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 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 caseEdit

Consider the tensor product of modules Hi(X; Z) ⊗ A. The theorem states there is a short exact sequence involving the Tor functor

 

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 cohomologyEdit

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

 

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

In fact, suppose

 

and define:

 

Then h above is the canonical map:

 

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 spaceEdit

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:

 

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

 

In fact the total cohomology ring structure is

 

CorollariesEdit

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.

 

where βi(X) are the Betti numbers of X and   is the torsion part of  . One may check that

 

and

 

This gives the following statement for integral cohomology:

 

For X an orientable, closed, and connected n-manifold, this corollary coupled with Poincaré duality gives that βi(X) = βni(X).

NotesEdit

ReferencesEdit

  • 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. S2CID 122894881.

External linksEdit