Groupe fondamental de l’espace des feuilles dans les feuilletages sans holonomie. (Fundamental group of the leaf space of foliations without holonomy). (French) Zbl 0714.57016

Levitt, Gilbert

J. Differ. Geom. 31, No. 3, 711-761 (1990).

The author studies \(C^ 2\)-foliations of codimension 1 with trivial holonomy in two cases: nonsingular foliations of noncompact manifolds, singular foliations of closed manifolds. The principal tool is a quotient of \(\pi_ 1M\)- the fundamental group of the leafspace \(\pi_ 1(M/{\mathcal F})\). Any subgroup of finite type of \(\pi_ 1(M/{\mathcal F})\) is a free product of abelian free groups. A geometric interpretation of the factors of rank \(\geq 2\) is given. In particular, the author deduces the absence of exceptional leaves under certain assumptions on \(\pi_ 1M\) (for example: \(\pi_ 1M\) is of finite type and has no free nonabelian quotient). These results are applied to the transversely affine foliations.

Reviewer: A.Piatkowski

MSC:

57R30

Foliations in differential topology; geometric theory

Keywords:

C\({}^ 2\)-foliations of codimension 1 with trivial holonomy; singular foliations of closed manifolds; fundamental group of the leafspace; exceptional leaves; transversely affine foliations