# The Stratified Spaces of Real Polynomials & Trajectory Spaces of Traversing Flows

@article{Katz2014TheSS,
title={The Stratified Spaces of Real Polynomials \& Trajectory Spaces of Traversing Flows},
author={Gabriel Katz},
journal={arXiv: Geometric Topology},
year={2014}
}
• G. Katz
• Published 10 July 2014
• Mathematics
• arXiv: Geometric Topology
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 \cite{K2}, for \emph{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 $\Omega^\bullet_{'\langle n]}$ which…

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## References

SHOWING 1-2 OF 2 REFERENCES

As has been observed by Morse [1], any generic vector field v on a compact smooth manifold X with boundary gives rise to a stratification of the boundary by compact submanifolds , where . Our main
Let $X$ be a compact smooth manifold with boundary. In this article, we study the spaces $\mathcal V^\dagger(X)$ and $\mathcal V^\ddagger(X)$ of so called boundary generic and traversally generic