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One-sided trivial-holonomy classes form a normal subgroup

Statement

Assume Countable Choice ${A C}_{ω}$ . Let $F$ be a transversely oriented codimension-one foliation, $L$ a leaf, $x \in L$ , and $j$ one of its two sides. The classes in $π_{1} (L, x)$ whose one-sided holonomy germ is the identity form a normal subgroup $N_{j} (L, x) ◃ π_{1} (L, x)$ . Consequently the quotient $P_{j} (L, x) = π_{1} (L, x) / N_{j} (L, x)$ used to define ordinary one-sided limit cycles is a group.

Facts & Assumptions

Given: A transversely oriented codimension-one foliation $F$ , a leaf $L$ , a base point $x \in L$ , one side $j$ of $L$ , and the standing countable choice assumption.

Proof

technique · direct

1.1given

By the definition of the holonomy representation, each class in $π_{1} (L, x)$ is assigned the germ of the return map along its reversed representative, on the chosen side (the library homomorphism convention), and the transverse orientation makes these germs side-preserving, so the assignment is a group homomorphism from $π_{1} (L, x)$ to the group of side-preserving germs of local diffeomorphisms of a half-transversal of the given side (The holonomy representation and the holonomy group of a leaf, Germs of local diffeomorphisms at a point).

2.1step 1.1∎

The classes in $π_{1} (L, x)$ whose one-sided holonomy germ is the identity are exactly the kernel of that homomorphism; the kernel of a group homomorphism is a normal subgroup, so the indicated classes form $N_{j} (L, x) ◃ π_{1} (L, x)$ , and the quotient $P_{j} (L, x) = π_{1} (L, x) / N_{j} (L, x)$ used to define ordinary one-sided limit cycles is a group; no choice principle beyond the standing assumption is used.

Depends on

Used by

Limit cycles of a leaf Definition

Dependency tree · two levels

23 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

S. P. Novikov, The Topology of Foliations (English translation by J. A. Zilber; complete PDF of the translation) (standard reference, not scraped)