Session S14 - Global Injectivity, Jacobian Conjecture, and Related Topics

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Topological classification of submersion functions

Ingrid S. Meza-Sarmiento

Universidade Federal de São Carlos, Brazil   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

Let $f, g : \mathbb{R}^2\to \mathbb{R}$ be two submersion functions and $\mathscr{F}(f)$ and $\mathscr{F}(g)$ be the regular foliations of $\mathbb{R}^2$ whose leaves are the connected components of the level sets of $f$ and $g$, respectively. We say that $f$ and $g$ are topologically equivalent (resp. o-topologically equivalent) if there exist homeomorphisms (resp. orientation preserving homeomorphisms) $h: \mathbb{R}^2\to \mathbb{R}^2$ and $\ell: \mathbb{R} \to \mathbb{R}$ such that $$ \ell \circ f = g \circ h. $$ The topological equivalence of $f$ and $g$ guarantees the topological equivalence of $\mathscr{F}(f)$ and $\mathscr{F}(g)$, but the converse is not true, in general. The main objective in this poster is to introduce a class of submersions, wide enough in order to contain non-trivial behaviors, Broughton example included, and to give necessary and sufficient conditions for that the converse implication be also valid inside this class. We also present a complete topological invariant for this class of submersions.

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