Homology of Spaces of Homogeneous Polynomials in R without Multiple Zeros

For any natural d ≥ k ≥ 2 we calculate the cohomology groups of the space of homogeneous polynomials R → R of degree d, which do not vanish with multiplicity ≥ k on real lines. For k = 2 this problem provides the simplest example of the situation, when the “finite-order” invariants of nonsingular objects are not a complete system of invariants. The “affine” version of this problem (the calculation of the homology group of the space of polynomials R → R with leading term x and without ≥ k-fold roots) was solved by V. I. Arnold in [2], see also [3]. As in these works, our present calculation is based on the study of the discriminant set, i.e. of the set of polynomials with forbidden multiple zeros. The problem solved below turns out to be more complicated, because an essential contribution to the homology group comes from the neighborhood of the “infinitely degenerate” polynomial equal identically to 0. By this reason, the method of simplicial resolutions of the discriminant set, solving immediately the “affine” problem, is replaced by its continuous analog: conical resolution, used previously in [5]. Here we have the simplest situation, when the invariants of “finite order” of the space of nonsingular objects do not constitute a complete system of invariants. Indeed, our spaces of nonsingular polynomials can be considered as finite-dimensional approximations of the space F \Σk of smooth functions S → R without k-fold zeros (if d is odd, then with values in a nontrivial line bundle); here S is realized as a half of the unit circle in R. By analogy with [4], the cohomology classes of “finite order” of the space F \Σk are exactly those, which are obtained by a natural stabilization of such cohomology groups for approximating spaces. It turns out, that for k = 2 and even d all such 0-dimensional