Is homology determined by cohomology?

Solution 1:

Suppose that $X$ is a space such that all of its homology groups are finitely generated. This holds, for example, if $X$ is a "levelwise finite" CW complex (finitely many cells in each dimension), which is a very broad class of spaces. Then universal coefficients implies that all of the cohomology groups are also finitely generated. Moreover, the isomorphism class of each homology and cohomology group is determined by rank and torsion subgroup, and universal coefficients implies that

• $H^k(X, \mathbb{Z})$ and $H_k(X, \mathbb{Z})$ have the same rank, and
• The torsion subgroup of $H^k(X, \mathbb{Z})$ is isomorphic to the torsion subgroup of $H_{k-1}(X, \mathbb{Z})$.

Hence the two sequences of isomorphism classes of groups determine each other in this case. They are nearly the same, except that the torsion subgroups are shifted one degree.

Without the finitely generated hypothesis this is no longer true. For every abelian group $A$ and positive integer $n \ge 1$ it is possible to construct a Moore space $X = M(A, n)$, which is a space whose homology vanishes except in degree $n$, where it is isomorphic to $A$, and in degree $0$, where it is $\mathbb{Z}$. By universal coefficients, the cohomology of a Moore space is

• $H^0(X, \mathbb{Z}) \cong \mathbb{Z}$
• $H^n(X, \mathbb{Z}) \cong \text{Hom}(A, \mathbb{Z})$
• $H^{n+1}(X, \mathbb{Z}) \cong \text{Ext}^1(A, \mathbb{Z})$

and all other cohomology vanishes. So the question is whether an abelian group $A$ is determined up to isomorphism by the isomorphism class of $\text{Hom}(A, \mathbb{Z})$ and $\text{Ext}^1(A, \mathbb{Z})$, and the answer is no. Counterexamples cannot be finitely generated: the first one that comes to mind is the following. We have

$$\text{Hom}(\mathbb{Q}, \mathbb{Z}) = 0$$

(straightforward) and

$$\text{Ext}^1(\mathbb{Q}, \mathbb{Z}) \cong \mathbb{R}$$

(nontrivial), and $\text{Ext}^1(-, -)$ preserves direct sums in the first factor, so it follows that the groups $\mathbb{Q}$ and $\mathbb{Q} \oplus \mathbb{Q}$ cannot be distinguished this way, since the two groups are not isomorphic, but $\mathbb{R} \cong \mathbb{R} \oplus \mathbb{R}$ (as abelian groups).