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Two nonhomotopic leaf loops can have the same holonomy germ

Statement refuted

Assume Countable Choice ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )). The following inference is false: two leaf loops with the same holonomy germ are leafwise homotopic relative to endpoints, so that the holonomy groupoid would coincide with the monodromy groupoid.

Counterexample. In the Möbius band foliation constructed below, the middle leaf has $π_{1} (L_{m i d}) ≅ Z$ generated by the core loop $γ$ , whose holonomy germ is the reflection $x \mapsto - x$ . The double $2 γ$ is not null-homotopic in the leaf, yet its holonomy germ is the square of the reflection, that is, the identity germ, as is the germ of the constant loop. Hence the constant loop and $2 γ$ are nonhomotopic relative to endpoints and have the same holonomy germ, and the kernel of the holonomy representation is the nontrivial subgroup $2 Z \leq Z$ ; the natural map $Mon (F) \to Hol (F)$ is not injective in general.

Facts & Assumptions

Given: The Möbius band foliation $F$ of $M = M o b$ with middle leaf $L_{m i d} ≅ S^{1}$ , its core loop $γ$ and its double $2 γ$ , and the constant loop $e$ at the base point.

[F1]

$π_{1} (L_{m i d}) ≅ Z$ , generated by the core loop $γ$ ; the holonomy representation sends $γ$ to the reflection germ $r : x \mapsto - x$ , and $r^{2} = i d$ (computed below).

[F2]

The holonomy group at a point is the image of the holonomy representation and is the isotropy group of the holonomy groupoid at that point (The isotropy of the holonomy groupoid is the leaf holonomy group, The holonomy representation and the holonomy group of a leaf).

[F3]

Holonomy germs are multiplicative under concatenation: $h_{a * b} = h_{b} \circ h_{a}$ , so the germ of $k$ times a loop is the $k$ -th power of its germ; the constant loop has the identity germ and represents the zero class (Holonomy respects path concatenation and reversal, Based loops and the fundamental group).

[F4]

Two loops are homotopic relative to endpoints exactly when they have the same class in $π_{1}$ ; in $Z$ the class of $2 γ$ is twice a generator, hence nonzero, while the constant loop represents $0$ (Based loops and the fundamental group, the local calculation below).

[F5]

The monodromy groupoid has arrows the leafwise homotopy classes of leafwise paths and the holonomy groupoid has arrows the holonomy classes; the projection sends a homotopy class to its holonomy class (The monodromy groupoid of a foliation, The holonomy groupoid of a foliation).

Counterexample

1.1givenconstructalgebra

Define the band as $(R \times (- 1, 1)) / Z$ with $k \cdot (t, x) = (t + k, (- 1)^{k} x)$ and foliation by horizontal lines. The middle leaf is $R / Z$ with fundamental group $Z$ by $Deg : π_{1} (R / Z, [0]) \to (Z, +)$ is an isomorphism (the core loop lifts from $0$ to $1$ and has degree one). Transport once around its generator holds $x$ constant on the covering strip and then uses the gluing $(1, x) \sim (0, - x)$ ; hence its return germ is $x \mapsto - x$ , a nonidentity involution. The suspension holonomy formula gives the same germ (the inverse reflection equals itself), and the $k$ -fold loop has germ $x \mapsto (- 1)^{k} x$ .

1.2F1F3

Same germ. By [F1] the class $[γ] \in π_{1} (L_{m i d})$ satisfies $ρ ([γ]) = r$ with $r^{2} = i d$ . By [F3] the double loop $2 γ$ has holonomy germ $r \circ r = i d$ , which is also the holonomy germ of the constant loop $e$ . Hence $e$ and $2 γ$ have the same holonomy germ.

1.3F1F4

Different homotopy classes. By [F4] and [F1], the class of $2 γ$ in $π_{1} (L_{m i d}) ≅ Z$ is twice a generator and hence nonzero, whereas the constant loop represents $0$ ; so $e$ and $2 γ$ are not leafwise homotopic relative to endpoints.

2.1F2F5step 1.2step 1.3

The kernel is nontrivial. The kernel of $ρ$ contains the class of $2 γ \neq 0$ , so $\ker ρ$ is a nontrivial subgroup of $Z$ , namely $2 Z$ . Consequently the holonomy group is $Z / 2$ , and the isotropy of the holonomy groupoid at the base point is $Z / 2$ while the corresponding isotropy of the monodromy groupoid is $Z$ ; the projection $Mon (F) \to Hol (F)$ identifies the two loops of steps 1.1 and 1.2 and is therefore not injective.

3.1step 1.2step 1.3step 2.1∎

Conclusion. Steps 1.1 and 1.2 exhibit two leaf loops that are nonhomotopic relative to endpoints yet have the same holonomy germ, refuting the stated inference; step 2.1 shows that the monodromy and holonomy groupoids are genuinely different quotients of the leafwise path groupoid.

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Sources

Leiden NCG seminar, Noncommutative Geometry of Foliations (2023 seminar notes) (standard reference, not scraped)
Eckhard Meinrenken, Lie Groupoids and Lie Algebroids, lecture notes (University of Toronto MAT1341, Fall 2017) (standard reference, not scraped)