homology over fields

Is it true that the homology of a manifold with field coefficients determines the homology
over the integers? I know that by the universal coefficient theorem that
$H_k(X; \mathbb{F}) \cong H_k(X; \mathbb{Z}) \bigotimes \mathbb{F}$ since the Tor part vanishes for fields. Then I think we can use the structure theorem of abelian groups (since $H_k(X; \mathbb{Z})$ is an abelian group). Is this true? Is there a reference for this claim?

Solutions Collecting From Web of "homology over fields"

No. More generally, let $X$ be a space whose integral homology is levelwise finitely generated. This is true for any compact manifold but not true for general noncompact manifolds, e.g. a surface of infinite genus. Then the integral homology groups $H_k(X, \mathbb{Z})$ are determined by their rank and by the number of times the summand $\mathbb{Z}_{p^k}$ shows up in their torsion subgroups.

The rank, and only the rank, can be detected by taking homology with coefficients in $\mathbb{Q}$, or more generally any field of characteristic $0$. Some information about the $p$-torsion can be detected by taking homology with coefficients in $\mathbb{F}_p$, or more generally any field of characteristic $p$, although less directly than you suggest: universal coefficients gives a short exact sequence

$$0 \to H_i(X, \mathbb{Z}) \otimes \mathbb{F}_p \to H_i(X, \mathbb{F}_p) \to \text{Tor}(H_{i-1}(X, \mathbb{Z}), \mathbb{F}_p) \to 0.$$

The Tor term here does not vanish in general; its rank is the rank of the $p$-torsion in $H_{i-1}(X, \mathbb{F}_p)$. The rank of the leftmost term is the rank of $H_i(X, \mathbb{Z})$ plus the sum of the number of times $\mathbb{Z}_{p^k}$ shows up over all $k$.

So the problem is that taking homology with coefficients in a field cannot tell you all of the information in the torsion: in particular it cannot distinguish between an integral homology group with torsion of the form $\mathbb{Z}_{p^2}$ and torsion of the form $\mathbb{Z}_{p^3}$, for example (and manifolds with torsion of this form are not hard to write down, e.g. using lens spaces). You can detect this by taking homology with coefficients in $\mathbb{Z}_{p^k}$ for all $k$, or more cleanly by taking homology with coefficients in the $p$-adic integers, which to avoid collision with my previous notation I’ll write $\widehat{\mathbb{Z}}_p$. A keyword to look up here is fracture theorem.