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A nontransverse pullback need not reproduce the rank of a foliation

Statement refuted

Assume Countable Choice ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )). The following inference is false: for an arbitrary smooth map $f : N \to M$ and a regular foliation $F$ of $M$ , the inverse-image spaces $D_{t}^{*} : = (d f_{t})^{- 1} (T_{f (t)} F)$ form a smooth distribution of constant rank $\dim N - codim F$ and define a regular pullback foliation whose leaves are the connected components of leaf preimages.

Counterexample. Let $M = R^{2}$ with the horizontal foliation, let $N = R$ , and let $f (t) = (0, t^{2})$ . Since $d f_{t} (v) = (0, 2 t v)$ and $T_{f (t)} F = R \times {0}$ , one has $D_{t}^{*} = {0}$ for $t \neq 0$ but $D_{0}^{*} = T_{0} R$ . Thus the rank jumps from $0$ to $1$ at $0$ , so $D^{*}$ is not a regular distribution. The map is transverse to $F$ for $t \neq 0$ and nontransverse at $0$ . This refutes the rank and regular-distribution inference without the transversality hypothesis.

Facts & Assumptions

Given: The horizontal foliation $F$ of $R^{2}$ , the map $f : R \to R^{2}$ , $f (t) = (0, t^{2})$ , and the family of inverse-image spaces $D_{t}^{*} = (d f_{t})^{- 1} (T_{f (t)} F)$ .

[F1]

The horizontal foliation $F$ of $R^{2}$ is the regular foliation whose leaves are the lines $R \times {c}$ ; its tangent distribution is $T_{f (t)} F = R \times {0}$ at every point, a rank-one subbundle of $T R^{2}$ (Flat charts for a distribution, Integrable distributions).

[F2]

A smooth map $f$ is transverse to $F$ at $t$ exactly when $d f_{t} (T_{t} N) + T_{f (t)} F = T_{f (t)} M$ , and the inverse-image convention for the pullback is $D_{t}^{*} = (d f_{t})^{- 1} (T_{f (t)} F)$ (Smooth maps transverse to a regular foliation, The pullback foliation under a transverse map).

[F3]

For smooth $f$ the differential $d f_{t}$ is the linear map of tangent spaces induced by $f$ , computed in coordinates by the Jacobian matrix (The differential of a smooth map).

[F4]

A smooth distribution of rank $k$ assigns to every point a $k$ -dimensional subspace as a smooth vector subbundle, so its rank is constant; the linear preimage of a linear subspace under a linear map is a linear subspace (Vector subbundles).

Counterexample

1.1F3given

In coordinates on $R^{2}$ and $R$ the map $f (t) = (0, t^{2})$ has Jacobian $(0, 2 t)^{T}$ , so $d f_{t} (v) = (0, 2 t v)$ for every $v \in T_{t} R$ , by [F3].

2.1F1F2step 1.1algebra

Hence $d f_{t} (v) \in T_{f (t)} F = R \times {0}$ holds exactly when $2 t v = 0$ . Therefore $D_{t}^{*} = {v : 2 t v = 0}$ equals ${0}$ for $t \neq 0$ and equals $T_{0} R$ at $t = 0$ .

2.2F1F2step 1.1

The map is transverse to $F$ for $t \neq 0$ : there $d f_{t} (T_{t} R)$ is the vertical line ${0} \times R$ , which together with $T_{f (t)} F = R \times {0}$ spans $T_{f (t)} R^{2}$ . At $t = 0$ the differential vanishes, so $d f_{0} (T_{0} R) = {0}$ and the sum $d f_{0} (T_{0} R) + T_{f (0)} F = R \times {0}$ is a proper subspace of $T_{f (0)} R^{2}$ : the map is not transverse at $0$ . Thus the failure of the rank conclusion occurs exactly at the point where transversality fails, and the transversality hypothesis of The pullback foliation under a transverse map is essential.

3.1F4step 2.1

The rank of $D_{t}^{*}$ is $0$ for $t \neq 0$ and $1$ at $t = 0$ ; in particular $D^{*}$ is not a smooth distribution of constant rank $\dim N - codim F = 1 - 1 = 0$ , and it is not a vector subbundle of $T R$ near $0$ . So the first two conclusions of the inference fail.

4.1F1F2step 2.1step 3.1∎

Finally the horizontal leaf $L_{c} = R \times {c}$ has preimage $f^{- 1} (L_{c}) = {t : t^{2} = c}$ , a finite set or empty; its connected components are points, whose tangent spaces are ${0}$ , while $D_{0}^{*} = T_{0} R$ . So the leaf-preimage description is not compatible with the inverse-image spaces at $0$ either, and the inference is false in every one of its clauses. The claim is refuted.

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Sources

Leiden NCG seminar, Noncommutative Geometry of Foliations (2023 seminar notes, complete 53-page PDF) (standard reference, not scraped)
Danny Calegari, Foliations and the Geometry of 3-Manifolds (Oxford Mathematical Monographs; author-hosted PDF) (standard reference, not scraped)