A UNIVERSAL COEFFICIENT THEOREM FOR COHOMOLOGY

HALVARD FAUSK

Abstract. Let X be any topological space. We present a universal coefficient type theorem expressing the integral homology H_{n}(X) as a subgroup of an extension A_{n}(X) of hom(H^{n}(X), ℤ) by Ext(H^{n+1}(X), ℤ). We describe the quotient group A_{n}(X)/H_{n}(X).

It is known that if the n and n + 1st singular cohomology groups of a topological space X vanish, then the nth singular homology group of the space X vanish. This is a consequence of Nunke’s theorem and the universal coefficient theorem which express the cohomology in terms of the homology. Nunke’s theorem says that if hom(A, ℤ) = 0 and Ext(A, ℤ) = 0 for an abelian group A, then A = 0. This indicates that there might be a universal coefficient theorem expressing the singular homology in terms of the singular cohomology [1]. The purpose of this note is to present such a theorem.

The dual of ⊕_{χ}ℤ is canonically isomorphic to ∏_{χ} ℤ. It is an amazing fact, sometimes called the Specker phenomena after its discoverer, that the dual of a product ∏_{χ} ℤ is canonically isomorphic to ⊕_{χ}ℤ for any non-measurable cardinality χ. If there are cardinalities that are not non-measurable they have to be very large.

Proposition 0.1 (Specker). The double dual map

ρ : ⨁_{χ} ℤ → hom(∏_{χ} ℤ, ℤ)

is an isomorphism for any non-measurable cardinality χ.

For a proof of this see [4, 5].

Let S_{•} be a chain complex of free abelian groups. We denote the chain complex hom(S_{•}, ℤ) by P^{•}. Note that the grading is such that P^{n} = hom(S_{n}, ℤ), and that each P^{n} is isomorphic to ∏_{A} ℤ where S_{n} = ⊕_{A}ℤ.

By Proposition 0.1 we get an isomorphism

S_{•} ≅ hom(P^{•}, ℤ).

The main example we have in mind is S_{•}(X, A) the singular chain complex. We have that P^{•} is the singular cochain complex

sing^{•}(X, A) = hom(sing_{•}(X, A), ℤ).

Date: March 1, 2003.

1991 Mathematics Subject Classification. Primary; Secondary .

Definition 0.2. Define the Ext homology with coefficients in M of S_{•} to be the homology of the complex

Ext(P^{•}, M).

We denote the Ext homology by ExH(S_{•}).

It is easy to see that a short exact sequence of chain complexes of free abelian groups gives a long exact sequence of Ext-homology. Ext homology of a finite sum of chain complexes is the sum of the homology of the individual complexes. For arbitrary sums of complexes this is false.

Proposition 0.3. Let S_{•} be a chain complex of free abelian groups. Then we have the following diagram where the vertical and horizontal sequences are short exact.

\xymatrix{ & H_{n}(S_{•}, ℤ)) \ar[d] \\ Ext(H^{n+1}(P^{•}), ℤ) \ar[r] & A_{n} \ar[r] \ar[d] & hom(H^{n}(P^{•}), ℤ) \\  & ExH_{n+1}(P^{•}).}

Moreover this diagram is natural in S_{•}.

Proof. Let

M → I_{0} → I_{1} → 0

be a injective resolution for M. The double complex spectral sequence of the double complex

hom(P^{•}, I_{•})

and Proposition 0.1 gives the two short exact sequences. This is also easily seen directly by using appropriate subcomplexes of the double complex and the long exact sequence in homology. □

Probably neither of the sequences splits in general.

The group Ext(∏ ℤ, ℤ) is a ℚ/ℤ-module. This is seen by using the injective resolution

ℤ → ℚ → ℚ/ℤ

of the integers Z. Note that hom(A, ℤ) is torsion free, hence free if A is finitely generated. We get that

Ext(H^{n+1}(S_{•}), ℤ) → H_{n}(S_{•}) → hom(H^{n}(S_{•}), ℤ)

when both the n and n + 1 cohomology groups are finitely generated.

Theorem 0.4. Let (X, A) be any pair of spaces. There are two short exact sequences

\xymatrix{ & H_{n}(X, A) \ar[d] \\ Ext(H^{n+1}(X, A), ℤ) \ar[r] & A_{n} \ar[r] \ar[d] & hom(H^{n}(X, A), ℤ) \\  & ExH_{n+1}(X, A).}

Note that we do not claim that any of the short exact sequences split.

Corollary 0.5. If f : (X, A) → (Y, B) is a cohomology isomorphism in degrees n and n + 1, then we get by the naturality of the diagram (and Nunke’s theorem [1]) that f induces an isomorphism in both homology and Ext-homology in degree n. We also have that if the cohomology vanish in degrees n and n + 1, then the homology and the Ext-homology vanish in degree n.

Remark 0.6. Torsion free abelian groups M for which the double dual map

ρ : ⊕_{ℕ}M → hom(∏_{ℕ}ℤ, M)

is an isomorphism are called slender groups. This implies that the double dual also is an isomorphism for all direct sums of ℤ over any non-measurable cardinal.

Here is a characterization of slender groups due to Nunke. A torsion free abelian group A is slender if and only if A does not contain the p-adic integers ℤp̂ or ℤ/p^{∞} = ℤ[\frac{1}{p}]/ℤ or ℚ for any prime p. The statement of this note generalizes to coefficients in any slender group M instead of the integers.

References

[1] Bredon G.E., On the Künneth formula and functorial dependence in algebraic topology, Amer. j. Math. 90 1968 522-527.

[2] R.J. Nunke and J.J. Rotman, Singular cohomology groups, J. London Math. Soc. 37 (1962), 301-306.

[3] R.J. Nunke, Slender groups, Acta Sci. Math. (Szeged) 23 1962 6773.

[4] E. Specker, Additive Gruppen von Folgen ganzer Zahlen, Portugalian Math. 9 (1950) 131-140.

[5] E.C. Zeeman, On direct sums of free cycles, J. London Math. Soc. 30 (1955), 195-212.

Department of Mathematics, Northwestern University, Evanston IL 60208

   E-mail address: fausk@math.northwestern.edu