Poincaré maps and Bol’s theorem
The usual Poincaré map is deﬁned for a four-web admitting three independent
Abelian relations, and gives a natural projective model of the web where the leaves
are straight lines. This model thus proves that the web is associated with a quartic
curve in the dual projective plane.
We shall present a generalization of this situation to the case of ﬁve-webs for
which every extracted three-web carries an Abelian relation. This generalization takes
into account not only the Abelian relations of the web, but also the spaces of Abelian
relations of the extracted sub-webs. This way we prove the following theorem of
If all extracted three-webs of a ﬁve-web admit an Abelian relation,
then the web is diffeomorphic either to ﬁve pencils of straight lines,
or to Bol’s exceptional example.
The main tool used in the separation of cases is a careful investigation of the
conﬁguration formed by the spaces of relations of the extracted sub-webs in the space
of relations of the whole web. (PDF ﬁle)