In algebraic topology, universal coefficient theorems (UCT) 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 :

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.

, 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 with coefficients in a field. These can differ, but only when the characteristic is a prime number-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

Furthermore, this sequence splits.

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

Universal coefficient theorem for cohomology

be a module over a principal ideal, or any field.)

There is 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

above is the canonical map:

An alternative point of view can be based on representing cohomology via Eilenberg–MacLane space takes a homotopy 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

, the real projective space.

Knowing that the integer homology is given by:

, so that the above exact sequences yield

. In fact the total cohomology ring structure is

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.

are the Betti numbers. One may check that

and

This gives the following statement for integral cohomology:

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

Universal coefficient spectral sequence

There is a generalization of the universal coefficient theorem for (co)homology with twisted coefficients.

For cohomology we have

is the Ext group.

Similarly for homology,

the Tor group.

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 the author's homepage.

• Jerome Levine. “Knot Modules. I.” Transactions of the American Mathematical Society 229 (1977): 1–50. https://doi.org/10.2307/1998498

External links

• Universal coefficient theorem with ring coefficients