2014/07/10 by Gabriel Katz, Katz, Gabriel
Engineering · Mathematics · #57R25 #57R35 #Advanced Differential Equations and Dynamical Systems #Advanced Numerical Analysis Techniques #FOS: Mathematics #Geometric Topology (math.GT) #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.1407.2984
openalex publication_date 2014/07/10 · openalex created_date 2024/04/11 · openalex updated_date 2026/07/28
This paper is the third in a series that researches the Morse Theory, gradient flows, concavity and complexity on smooth compact manifolds with boundary. Employing the local analytic models from \citeK2, for traversally generic flows on (n+1)-manifolds X, we embark on a detailed and somewhat tedious study of universal combinatorics of their tangency patterns with respect to the boundary \d X. This combinatorics is captured by a universal poset Ω^\bullet'⟨ n] which depends only on the dimension of X. It is intimately linked with the combinatorial patterns of real divisors of real polynomials in one variable of degrees which do not exceed 2(n+1). Such patterns are elements of another natural poset Ω⟨ 2n+2] that describes the ways in which the real roots merge, divide, appear, and disappear under deformations of real polynomials. The space of real degree d polynomials \mathcal Pd is stratified so that its pure strata are cells, labelled by the elements of the poset Ω⟨ d]. This cellular structure in \mathcal Pd is interesting on its own right (see Theorem \ref th4.1 and Theorem \ref th4.2). Moreover, it helps to understand the localized structure of the trajectory spaces \mathcal T(v) for traversally generic fields v, the main subject of Theorem \ref th5.2 and Theorem \ref th5.3.