Correspondences in Topology

For simplicity and definiteness, let's assume that $X$ and $Y$ are smooth and compact (and orientable, which will always be the case if they are complex varieties), and let $n$ be the dimension of $Y$. First of all, it might help to note that $H^n(X\otimes Y) \cong H^\*(X)\otimes H^{n-\*}(Y) \cong Hom(H^\*(Y),H^\*(X))$, where for the final assertion I am using that $Y$ is smooth and compact, so that its cohomology satisfies Poincare duality. Thus if $Z$ is a cycle in $X\otimes Y$, of dimension equal to that of $X$ (and so of comdimension $n$) it induces a cycle class in $H^n(X\times Y)$, which in turn induces a map from cohomology of $X$ to that of $Y$. If $f:X\to Y$ and $Z = \Gamma_f$ is the graph of $f$ then this map is just the pull-back of cohomology classes by $f$.

So correspondences in the sense of physical cycles on $X\times Y$ induce correspondences in the sense of cohomology classes on $X\times Y$, which in turn induce morphisms on cohomology. If you like, you can strengthen the analogy with the $\Gamma_f$ case by thinking of a correspondence as a multi-valued function. Functions induce morphisms on cohomology; but since cohomology is linear (you can add cohomology classes), correspondences also induce morphisms on cohomology (you can simply add up the multiples values!). This gives the same construction as the more formal one given above.

Ben Webster notes in his answer that geometric representation theory provides a ready supply of correspondences. So does the theory of arithmetic lattices in Lie groups and the associated symmetric space. (I am thinking of Hecke correspondences and the resulting action of Hecke operators on cohomology.)

A very general framework, which I think covers both contexts, is as follows: suppose that a group $G$ acts on a space $X$, and that $H \subset Aut(X)$ is another subgroup commensurable with $G$, i.e. such that $G \cap H$ has finite index in each of $G$ and $H$.

Then (perhaps under some mild assumptions) $(G\cap H)\backslash X \hookrightarrow (G\backslash X \times H\backslash X) $ is a correspondence (in the physical, geometric sense) which will give a correspondence in cohomology via its cycle class. The resulting maps on cohomology are (a very general form of) Hecke operators.

Correspondences come into their own in situations where there aren't enough maps for functorial methods to be useful.

In symplectic topology, the only structure-preserving maps are symplectic immersions, and of those, the only ones which play well with the main technique of the subject - holomorphic curves - are the symplectomorphisms. So usually one treats a symplectic manifold in splendid isolation.

But Lagrangian correspondences arise very naturally - as graphs of symplectomorphisms; via symmetry (moment maps); via degeneration (vanishing cycles); and via elliptic boundary problems relevant to low-dimensional topology. They make excellent boundary conditions for holomorphic curves. Ongoing work of Ma'u-Wehrheim-Woodward develops this idea to bring functorial methods to bear, for the first time, in symplectic topology.

1) The analogy is that a smooth subscheme of $X\times Y$ defines an element of the Borel-Moore homology of $X\times Y$ which in the compact case is the same as homology (which is dual to cohomology; of course there's no good reason to make the topological definition with cohomology instead of homology). More generally, this is a special case of the analogy between Chow groups and (co)homology, which is quite well developed in algebraic geometry.

2) Look in Chriss and Ginzburg's Complex Geometry and Representation Theory. Essentially the entire book is about correspondences in the topological sense. Correspondences are very important throughout geometric representation theory, since they have a multiplication, and thus can be used to produce algebras acting on the cohomology of spaces.