SPACES OF POLYNOMIALS WITH CONSTRAINED DIVISORS AS GRASSMANIANS FOR TRAVERSING FLOWS

Abstract

We study traversing (i.e., admitting a Lyapunov function) vector flow on a smooth compact manifold with boundary. Fix a compact manifold $\hat{X}$ and a traversing vector field $\hat{v}$ on it, where the boundary $\partial \hat{X}$ is convex with respect to $\hat{v}$. Consider submersions/embeddings $\alpha: X \rightarrow \hat{X}$ such that the compact manifolds $X, \hat{X}$ are equidimensional and $\alpha(\partial X)$ avoids a priory chosen combinatorial 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 the submersion $\alpha:(X, v) \rightarrow(\hat{X}, \hat{v})$ a convex 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 convex pair $(\hat{X}, \hat{v})$, we introduce an equivalence relation among submersions $\alpha$, which we call a quasitopy. The notion of quasitopy is a crossover between bordisms of envelops and their pseudo-isotopies. In the study of quasitopies $\mathcal{Q} \mathcal{T}_d(\hat{X}, \hat{v} ; \mathbf{c} \Theta)$, 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. Thus, we associate with each envelop $\alpha$ an element $[\alpha] \in \mathcal{Q} \mathcal{T}_d(\hat{X}, \hat{v} ; \mathbf{c} \Theta)$, which is an invariant under special cobordisms and pseudo-isotopies.

We compute in the homotopy-theoretical terms, which involve $(\hat{X}, \hat{v})$ and $\mathcal{P}_d^{\mathbf{c} \Theta}$, the quasitopies $\mathcal{Q} \mathcal{T}_d(\hat{X}, \hat{v} ; \mathbf{c} \Theta)$. Then we prove that the quasitopies $\mathcal{Q} \mathcal{T}_d(Y, \mathbf{c} \Theta)$ often stabilize, as $d \rightarrow \infty$.

Received: February 19, 2023 
Accepted: May 2, 2023