Spaces of polynomials with constrained divisors as Grassmanians for traversing flows

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 $α: X \to \hat X$ such that $\dim X = \dim \hat X$ and $α(\partial X)$ avoids a priory chosen tangency patterns $Θ$ to the $\hat v$-trajectories. In particular, for each $\hat v$-trajectory $\hatγ$, we restrict the cardinality of $\hatγ\cap α(\partial X)$ 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 $v = α^\dagger(\hat v)$ is the $α$-transfer of $\hat v$ to $X$. For a fixed $(\hat X, \hat v)$, we introduce an equivalence relation among convex pseudo-envelops/ envelops $α: (X, v) \to (\hat X, \hat v)$, 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 $\mathcal{QT}_d(Y, \mathbf cΘ)$, the spaces $\mathcal P_d^{\mathbf cΘ}$ of real univariate polynomials of degree $d$ with real divisors whose combinatorial types avoid the closed poset $Θ$ play the classical role of Grassmanians. We compute, in the homotopy-theoretical terms that involve $(\hat X, \hat v)$ and $\mathcal P_d^{\mathbf cΘ}$, the quasitopies of convex envelops which avoid the $Θ$-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 $\mathcal{QT}_d(Y, \mathbf cΘ)$ often stabilize, as $d \to \infty$.

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