Counting ramified converings and intersection theory on spaces of rational functions. I. Cohomology of Hurwitz spaces

The Hurwitz space is a compactification of the space of rational functions of a given degree. The Lyashko–Looijenga map assigns to a rational function the set of its critical values. It is known that the number of ramified coverings of $\mathbb CP^1$ by $\mathbb CP^1$ with prescribed ramification points and ramification types is related to the degree of the Lyashko–Looijenga map on various strata of the Hurwitz space. Here we explain how the degree of the Lyashko–Looijenga map is related to the intersection theory on this space. We describe the cohomology algebra of the Hurwitz space and prove several relations between the homology classes represented by various strata.