We study traversing (i.e., admitting a Lyapunov function) vector  flow on a smooth compact manifold  with boundary. Fix a compact manifold  and a traversing vector field  on it, where the boundary is convex with respect to  Consider submersions/embeddings  such that the compact manifolds  are equidimensional and  avoids a priory chosen combinatorial tangency patterns  to the -trajectories. In particular, for each ‑trajectory  we restrict the cardinality of  by an even number d. We call the submersion  a convex envelop of the pair  Here the vector field  is the ‑transfer of  to X.

For a fixed convex pair we introduce an equivalence relation among submersions α, 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  the spaces  of real univariate polynomials of degree d with real divisors, whose combinatorial types avoid the closed poset  play the classical role of Grassmanians. Thus, we associate with each envelop α an element  which is an invariant under special cobordisms and pseudo-isotopies.

We compute in the homotopy-theoretical terms, which involve  and  the quasitopies  Then we prove that the quasitopies often stabilize, as

traversing vector flow, quasitopy, Grassmanians.

Received: February 19, 2023; Accepted: May 2, 2023; Published: June 15, 2023

How to cite this article: Gabriel Katz, Spaces of polynomials with constrained divisors as Grassmanians for traversing flows, JP Journal of Geometry and Topology 29(1) (2023),     47-120. http://dx.doi.org/10.17654/0972415X23005

This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License

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