On some topological invariants of algebraic functions associated to the Young stratification of polynomials

The connections between braid groups, algebraic functions and spaces of generic polynomials have been described by Arnolʹd in his famous paper on topological invariants of algebraic functions. In this seminal paper Arnolʹd studied the cohomology rings of the spaces of polynomials without multiple roots and showed that these rings satisfy three theorems: the stabilization theorem (the rings stabilize has the degree of the polynomials tends to infinity), the repetition theorem (the rings associated to spaces of polynomials of successive even and odd degrees are isomorphic) and the finiteness theorem (all cohomology groups are finite except the first two). We generalize the results of Arnolʹd by considering cohomology rings dual to arbitrary Young strata of the space of polynomials. We show that the stabilization, repetition and finiteness theorems still hold mutatis mutandis in this more general context.