2022/02/02 by Gabriel Katz, Katz, Gabriel
Mathematics · #Advanced Differential Equations and Dynamical Systems #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.2202.00862
openalex publication_date 2022/02/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study \sf traversing vector flows v on smooth compact manifolds X with boundary. For a given compact manifold X, equipped with a traversing vector field v which is \sf convex with respect to ∂ X, we consider submersions/embeddings α: X → X such that dim X = dim X and α(∂ X) avoids a priory chosen tangency patterns Θ to the v-trajectories. In particular, for each v-trajectory γ, we restrict the cardinality of γ∩ α(∂ X) by an even number d. We call ( X, v) a \sf convex pseudo-envelop/envelop of the pair (X, v). Here the vector field v = α^†( v) is the α-transfer of v to X. For a fixed ( X, v), we introduce an equivalence relation among convex pseudo-envelops/ envelops α: (X, v) → ( X, 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 QTd(Y, \mathbf cΘ), the spaces \mathcal Pd\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 ( X, v) and \mathcal Pd\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 QTd(Y, \mathbf cΘ) often stabilize, as d → ∞.