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I assume that $H_n(X,\mathbb{Z})$ are finitely generated free ablelian groups. Then from the universal coefficients theorem I have $H^n(X,\mathbb{Z})=Hom(H_n(X,\mathbb{Z})).$

Then, since $H_n(X,\mathbb{Z})$ is just $\mathbb{Z}^m$, we have $Hom(H_n(X,\mathbb{Z}),\mathbb{Z})=H_n(X,\mathbb{Z})$ through dual basis. Thus, taking $Hom$ from both sides, we get $Hom(H^n(X,\mathbb{Z}),\mathbb{Z})=H_n(X,\mathbb{Z})$. Is it a legit argument?

However, if I start with no information on homologies, and all information on cohomologies (for example, let's assume that $\forall n~$ $H^n(X,\mathbb{Z})$ is finitely generated free abelian group), can I somehow deduce what $H_n(X,\mathbb{Z})$ looks like?

Haldot
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Suppose your $X$ has finitely many cells in each dimension. Then by the universal coefficient theorem, $H^{n}(X)=\operatorname{Hom}(H_{n}(X),\mathbb{Z}) \oplus \operatorname{Ext}(H_{n-1}(X), \mathbb{Z})$. Since we are finitely generated, we have a spitting into a free part and a torsion part. Since $\operatorname{Ext}(\mathbb{Z}/p^n,\mathbb{Z})=\mathbb{Z}/p^n$, and since any nontrivial homomorphism into $\mathbb{Z}$ has infinite order, we deduce that the rank of $H_{n}(X)$ is the rank of $H^{n}(X)$ and the torsion part of $H_{n}(X)$ is the torsion part of $H^{n+1}(X)$.

In general you cannot solve the problem because I believe you have groups with isomorphic sets of homomorphisms into the integers and ext groups with the integers. Then you get an example by the Moore spaces for these groups.

Connor Malin
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    I should link this https://math.stackexchange.com/questions/1268593/is-homology-determined-by-cohomology which gives the exact same answer as me, but actually gives an example of such a group. – Connor Malin May 20 '20 at 00:00
  • Thanks. But is what I wrote about the case when we know homologies true? I mean, that $H_n(X,\mathbb{Z})=Hom(H^n(X,\mathbb{Z}),\mathbb{Z})$ – Haldot May 20 '20 at 00:28
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    Yes, your reasoning is correct. It is the same as this proof but you remove all torsion – Connor Malin May 20 '20 at 01:06