Spaces of polynomials as Grassmanians for immersions and embeddings

Abstract

Let $Y$ be a smooth compact $n$-manifold. We study smooth embeddings and immersions $\beta: M \to \mathbb R \times Y$ of compact $n$-manifolds $M$ such that $\beta(M)$ avoids some a priory chosen closed poset $\Theta$ of {\sf tangent patterns} to the fibers of the obvious projection $\pi: \mathbb R \times Y \to Y$. Then, for a fixed $Y$, we introduce an equivalence relation between such $\beta$'s; it is a crossover between pseudo-isotopies and bordisms. We call this relation {\sf quasitopy}. In the study of quasitopies, the spaces $\mathcal P_d^{\mathbf c\Theta}$ of real univariate polynomials of degree $d$ with real divisors, whose combinatorial patterns avoid a given closed poset $\Theta$, play the classical role of Grassmanians. We compute the quasitopy classes $\mathcal{QT}_d^{\mathsf{emb}}(Y, \mathbf c\Theta)$ of $\Theta$-constrained embeddings $\beta$ in terms of homotopy/homology theory of spaces $Y$ and $\mathcal P_d^{\mathbf c\Theta}$. We prove also that the quasitopies of emeddings stabilize, as $d \to \infty$.