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".



Why is the dual of a torus the same as its fundamental group?
The two are naturally dual lattices. The fundamental group of a torus $T$ can be canonically identified with the group (known as the cocharacter lattice) of $\it homomorphisms$ from the circle group to $T$, or equivalently the kernel of the (universal cover=exponential map) homomorphism from the Lie algebra $t$ to $T$. The characters of the torus on the other hand can be identified naturally with a subsest of $t^\*$, i.e. characters of the Lie algebra which are integral on the kernel of $t\to T$, i.e. the dual to the cocharacters.


As for the general question about Lie groups it seems way too general (note that a connected group acts trivially on its cohomology groups, so the relation is not that..) One direction to read about is the relation between the cohomology of the group, that of its classifying space, group cohomology and invariant polynomials (or "Casimirs") on the Lie algebra.


