Spaces of polynomials with constrained divisors as Grassmanians for traversing flows

TL;DR

This paper introduces a new geometric framework called quasitopy for studying traversing flows on manifolds, connecting it with polynomial divisor spaces that serve as Grassmannian analogs, and establishes stability and invariants of these structures.

Contribution

It defines quasitopy as a novel equivalence relation for convex pseudo-envelops, linking it to polynomial divisor spaces and characteristic classes, and proves their stabilization as polynomial degree increases.

Findings

Quasitopy classes are characterized by new invariants.

Polynomial divisor spaces act as Grassmannian analogs in this context.

Quasitopies often stabilize as polynomial degree tends to infinity.

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 $v = \alpha^\dagger(\hat v)$ 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 $\mathcal{QT}_d(Y, \mathbf c\Theta)$, 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 $\mathcal{QT}_d(Y, \mathbf c\Theta)$ often stabilize, as $d \to \infty$.