The set of continuous homomorphisms from a torus ${\mathbb T}^n = ({\mathbb R}/{\mathbb Z})^n \to {\mathbb R}/{\mathbb Z}$ can be identified with ${\mathbb Z}^n$ if we assign to each $k = (k_1, \ldots k_n) \in {\mathbb Z}^n$ the character $x \mapsto k \cdot x$.

The fundamental group of homotopy classes of loops ${\mathbb R}/{\mathbb Z} \to {\mathbb T}$ can also be identified with ${\mathbb Z}^n$ because each equivalence class (with base point at the origin) can be represented by $x \mapsto (k_1 x, k_2 x, \ldots, k_n x)$ for some $k \in {\mathbb Z}^n$.

My question is basically how much of a coincidence this isomorphism is. For one thing, it can't be too natural because the fundamental group pushes forward under a map, whereas the character group pulls back. So the natural question should be whether there is a natural relationship at the level of the first cohomology group instead of the fundamental group.

Of course, totally disconnected groups have interesting dual groups even though their cohomology is uninteresting as far as I know. And ${\mathbb R}^n$, being isomorphic to its dual but contractible, does not seem to exhibit a similar relationship.

But for a Lie group, say, I would like to know if there's a natural relationship between its representation theory (e.g. irreducible representations) on the one hand and its topology (e.g. cohomology) on the other hand. It might be no deeper than "well, the cohomology groups are representations".