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arXiv · 1111.2364

Generic Pseudogroups on (C,0) and the Topology of Leaves

Jean-François Mattei·Julio C. Rebelo·Helena Reis
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Abstract

We show that generically a pseudogroup generated by holomorphic diffeomorphisms defined about $0 \in \mathbb{C}$ is free in the sense of pseudogroups even if the class of conjugacy of the generators is fixed. This result has a number of consequences on the topology of leaves for a (singular) holomorphic foliation defined on a neighborhood of an invariant curve. In particular in the classical and simplest case arising from local foliations possessing a unique separatrix that is given by a cusp of the form $\{y^2-x^{2k+1}=0\}$, our results allow us to settle the problem of showing that a generic foliation possesses only countably many non-simply connected leaves and that this countable set is, indeed, infinite.

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Jean-François Mattei, Julio C. Rebelo, Helena Reis. 2011-11-09. Generic Pseudogroups on (C,0) and the Topology of Leaves. https://arxiv.org/abs/1111.2364

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