Spaces of polynomials with constrained divisors as Grassmanians for traversing flows

Abstract

We study {\sf traversing} vector flows $v$ on smooth compact manifolds $X$ with boundary. For a given compact manifold $\hat{X}$, equipped with a traversing vector field $\hat{v}$ which is {\sf convex} with respect to $\partial \hat{X}$, we consider submersions/embeddings $\alpha :X\to \hat{X}$ such that $\dim X=\dim \hat{X}$ and $\alpha (\partial X)$ avoids a priory chosen tangency patterns $\Theta$ to the $\hat{v}$-trajectories. In particular, for each $\hat{v}$-trajectory $\hat{\gamma }$, we restrict the cardinality of $\hat{\gamma }\cap \alpha (\partial X)$ by an even number $d$. We call $(\hat{X},\hat{v})$ a {\sf convex pseudo-envelop/envelop} of the pair $(X,v)$. Here the vector field $v={\alpha }^{†}(\hat{v})$ is the $\alpha$-transfer of $\hat{v}$ to $X$. For a fixed $(\hat{X},\hat{v})$, we introduce an equivalence relation among convex pseudo-envelops/ envelops $\alpha :(X,v)\to (\hat{X},\hat{v})$, which we call a {\sf quasitopy}. The notion of quasitopy is a crossover between bordisms of pseudo-envelops and their pseudo-isotopies. In the study of quasitopies ${\mathcal{Q}\mathcal{T}}_d(Y,\mathbf{c}\Theta )$, the spaces ${\mathcal{P}}_d^{\mathbf{c}\Theta }$ of real univariate polynomials of degree $d$ with real divisors whose combinatorial types avoid the closed poset $\Theta$ play the classical role of Grassmanians. We compute, in the homotopy-theoretical terms that involve $(\hat{X},\hat{v})$ and ${\mathcal{P}}_d^{\mathbf{c}\Theta }$, the quasitopies of convex envelops which avoid the $\Theta$-tangency patterns. We introduce characteristic classes of pseudo-envelops and show that they are invariants of their quasitopy classes. Then we prove that the quasitopies ${\mathcal{Q}\mathcal{T}}_d(Y,\mathbf{c}\Theta )$ often stabilize, as $d\to \infty$.