I'm learning about de Rham's theorem, but from a very analytical point of view: my algebra is very weak.
My understanding is that de Rham's theorem tells us that for a smooth manifold $M$, the de Rham cohomology groups $H_{DR}^k(M)$ are isomorphic to the singular cohomology groups $H^k(M;\mathbb{R})$ with real coefficients.
From what I gather by looking at simple examples (sphere, torus) and also informal discussions of topological "holes," these real singular cohomology groups are also isomorphic to the corresponding real singular homology groups, $H_k(M;\mathbb{R})$. But I can't find a straightforward discussion or statement of this fact.
So I have two questions:
- Is it true that for a smooth manifold $M$ (or possibly any topological space $M$), $H^k(M;\mathbb{R})\cong H_k(M;\mathbb{R})$? If so, how is this proven?
Reading the page on cohomology on Wikipedia, I find the following sentence after a remark about the universal coefficient theorem: "A related statement is that for a field $F$, $H^i(X,F)$ is precisely the dual space of the vector space $H_i(X,F)$." If I could prove this stament, I could then answer my question in the affirmative: since $H_i(X,F)$ and $H^i(X,F)$ are finite-dimensional vector spaces and dual, they must be isomorphic (although not canonically so). But I don't know how to prove the quoted statement, and I'm not sure of its precise relationship to the universal coefficient theorem; Wikipedia merely says this fact is "related," not that it is a consequence of UCT.
- Does this fact descend to integral coefficients too, at least for smooth manifolds manifolds? That is, is it true that $H^k(M;\mathbb{Z})\cong H_k(M;\mathbb{Z})$? If not, what is a counterexample?
Edit. Regarding question 2, I've just noticed that real projective spaces provide counterexamples for the case of integral coefficients; thus the answer is no. But does the result hold if none of the groups have torsion?
Best Answer
When the coefficient ring $R$ is the integers or a field, as long as the homology groups of $X$ are free (automatic over a field) and finitely generated over the coefficient ring, then $H_k(X; R) \cong H^k(X; R)$, by the Universal Coefficient Theorem. Furthermore, if $X$ has the structure of a finite CW complex, for example if $X$ is a compact manifold, then the homology groups will be finitely generated. There is no nice topological description that ensures that the integral homology groups will be torsion-free; spaces with cells in only even dimensions have this property, for example, but that's not a very broad class of spaces.