My favorite connection in mathematics (and an interesting application to physics) is a simple corollary from Hodge's decomposition theorem, which states:

On a (compact and smooth) riemannian manifold $M$ with its Hodge-deRham-Laplace operator $\Delta,$ the space of $p$-forms $\Omega^p$ can be written as the orthogonal sum (relative to the $L^2$ product) $$\Omega^p = \Delta \Omega^p \oplus \cal H^p = d \Omega^{p-1} \oplus \delta \Omega^{p+1} \oplus \cal H^p,$$ where $\cal H^p$ are the harmonic $p$-forms, and $\delta$ is the adjoint of the exterior derivative $d$ (i.e. $\delta = \text{(some sign)} \star d\star$ and $\star$ is the Hodge star operator).
(The theorem follows from the fact, that $\Delta$ is a self-adjoint, elliptic differential operator of second order, and so it is Fredholm with index $0$.)

From this it is now easy to proof, that every not trivial deRham cohomology class $[\omega] \in H^p$ has a unique harmonic representative $\gamma \in \cal H^p$ with $[\omega] = [\gamma]$. Please note the equivalence $$\Delta \gamma = 0 \Leftrightarrow d \gamma = 0 \wedge \delta \gamma = 0.$$

Besides that this statement implies easy proofs for PoincarÃ© duality and what not, it motivates an interesting viewpoint on electro-dynamics:

Please be aware, that from now on we consider the Lorentzian manifold $M = \mathbb{R}^4$ equipped with the Minkowski metric (so $M$ is neither compact nor riemannian!). We are going to interpret $\mathbb{R}^4 = \mathbb{R} \times \mathbb{R}^3$ as a foliation of spacelike slices and the first coordinate as a time function $t$. So every point $(t,p)$ is a position $p$ in space $\mathbb{R}^3$ at the time $t \in \mathbb{R}$. Consider the lifeline $L \simeq \mathbb{R}$ of an electron in spacetime. Because the electron occupies a position which can't be occupied by anything else, we can remove $L$ from the spacetime $M$.

Though the theorem of Hodge does not hold for lorentzian manifolds in general, it holds for $M \setminus L \simeq \mathbb{R}^4 \setminus \mathbb{R}$. The only non vanishing cohomology space is $H^2$ with dimension $1$ (this statement has nothing to do with the metric on this space, it's pure topology - we just cut out the lifeline of the electron!). And there is an harmonic generator $F \in \Omega^2$ of $H^2$, that solves $$\Delta F = 0 \Leftrightarrow dF = 0 \wedge \delta F = 0.$$ But we can write every $2$-form $F$ as a unique decomposition $$F = E + B \wedge dt.$$ If we interpret $E$ as the classical electric field and $B$ as the magnetic field, than $d F = 0$ is equivalent to the first two Maxwell equations and $\delta F = 0$ to the last two.

So cutting out the lifeline of an electron gives you automagically the electro-magnetic field of the electron as a generator of the non-vanishing cohomology class.