Spaces of Polynomials as Grassmanians for Immersions and Embeddings

Let Y be a smooth compact n-manifold. We studied smooth embeddings and immersions β:M→R×Y of compact n-manifolds M such that β(M) avoids some priory chosen closed poset Θ of tangent patterns to the fibers of the obvious projection π:R×Y→Y. Then, for a fixed Y, we introduced an equivalence relation between such β’s; creating a crossover between pseudo-isotopies and bordisms. We called this relation quasitopy. In the presented study of quasitopies, the spaces PdcΘ of real univariate polynomials of degree d with real divisors, whose combinatorial patterns avoid a given closed poset Θ, play the classical role of Grassmanians. We computed the quasitopy classes Qdemb(Y,cΘ) of Θ-constrained embeddings β in terms of homotopy/homology theory of spaces Y and PdcΘ. We proved also that the quasitopies of embeddings stabilize, as d→∞.