I had one or two little fights with correspondences in the context of algebraic geometry where an elementary correspondence $C:X\to Y$ of connected smooth $k$-Schemes seems to be defined as an irreducible closed (reduced) subscheme $C\hookrightarrow Y\times_k X$, such that the projection to $X$ is finite an surjective.

Further, I have vaguely heard of correspondences in topology (invented by Lefschetz?) where it seems to me, that such a thing is a cohomology class in $H^*(Y\times X)$ for compact, oriented manifolds $X,Y$. Using Poincare duality and the cohomology pushforward functor $(-)_!$ I can associate a cohomology class $(\Delta_f)_!(1)$ in $H^*(Y\times X)$ to a map $X\to Y$.

My questions are:

1. I can not really see the analogy of the concepts, except that in booth cases one can associate a corresponces to a morphism by its graph. So, what (or how deep) is the analogy?

2. I know (only a very few) applications of correspondences in algebraic geometric, but of none in topology. What are they good for? Where can I find applications?