As mentioned in the previous answer, the Atiyah-Singer index theorem is an excellent answer to your question. I would like to convince you that, in a sense, it is probably the only answer to your question. Fortunately that one theorem admits so many applications, generalizations, and elaborations that it almost becomes an area of mathematics unto itself (particularly when infused with the tools of C*-algebra theory).
My first remark is that K-theory is an inherently global tool - its power lies in the fact that it is built from but insensitive to the details of local geometry. From what I understand about PDE theory many of the interesting questions live on open balls in Euclidean space, about which algebraic topology in general has little new to contribute. Even when one considers boundary value problems where the geometry gets a little more interesting, the challenges are usually local on the boundary (i.e. the concern is with smoothness, not interesting global structure).
Once we have accepted that we are looking for a global answer to your question, it is natural to ask: is there a sense in which PDE's organize themselves into a full-fledged (co)homology theory? This, after all, is the way that topology usually interacts with other parts of mathematics: one begins with objects whose structure one wants to globalize (e.g. embedded loops, differential forms, vector bundles...) and one aims to build algebraic invariants out of those objects. In the case of PDE's the answer is K-homology, a generalized homology theory for the category of manifolds with the property that every first order linear elliptic operator $D$ on a manifold $M$ gives rise to a class $[D]$ in $K_*(M)$. K-homology, as the name suggests, is the homology theory which is naturally dual to K-theory, regarded as a (generalized) cohomology theory on the category of manifolds.
So the question is: what can we do with K-homology? The answer is that one can do quite a lot, but as with many constructions in algebraic topology many of the most interesting results involve pairings between homology and cohomology. The most fundamental pairing between K-homology and K-theory is the so-called index pairing, which takes an elliptic operator and a vector bundle and spits out the Fredholm index of the operator "twisted" by the bundle. The Atiyah-Singer index theorem is really a theorem about the topological properties of this pairing, and consequently it plays a central role in applications of K-homology.