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A paired regular disk sweep is open across its base gluing

Statement

Let $D$ be an actual compact C² disk region homeomorphic to the closed disk, and let G:D×[a,b]→M be a C² immersed disk sweep, transverse in the time direction with one constant sign. Suppose an interior subdisk U⊂D at the b-base is identified to the whole a-base by a diffeomorphism h:D→U satisfying G(y,a)=G(h(y),b), with their leafwise tangent maps agreeing under h. On the quotient X, the induced map is locally open at every interior seam point; its compact image has boundary contained in the image of ∂X. No global injectivity is required.

Facts & Assumptions

Given: A $C^{2}$ immersed disk sweep $G : D \times [a, b] \to M$ transverse in the time direction with one constant sign, and a base identification $h : D \to U$ of an interior subdisk $U$ at the $b$ -base with the whole $a$ -base such that $G (y, a) = G (h (y), b)$ and the leafwise tangent maps agree under $h$ .

[F1]

The in-pair item The canonical Jordan cap bundle develops coherently over every positive band supplies coherent regular $C^{2}$ cap development with $V$ -orbit tracks. The base diffeomorphism $h$ and the equality of the two base maps and their leafwise differentials are separate hypotheses of this item, not conclusions of that supplier; the sibling-pair item lem-finite-chart-surface-normal-forms-supply-jordan-disks-and-torsion-free-groups supplies the leafwise disk and plaque structure used locally.

[F2]

A $C^{2}$ map with invertible derivative has a $C^{2}$ local inverse, and a $C^{2}$ scalar equation with nonzero normal derivative has a unique local $C^{2}$ root (C² inverses and scalar return roots).

[F3]

The standing assumption is Countable Choice ${A C}_{ω}$ as recorded for this pair (The countable-choice principle used in the foliation pair).

Proof

technique · direct

1.1F1F2givenconstruct

At a seam point choose a small leafwise plaque chart around its common image. The $a$ -base and $b$ -base maps have invertible leafwise differentials by the immersion and agreement hypotheses, so after shrinking each has a plaque inverse branch and its nearby time slices are graphs over that plaque patch by [F2]. The differential in the time direction has the same nonzero transverse sign on both sheets. The piece of the cylinder adjoining the $a$ -base has parameter $t > a$ , whereas the piece adjoining the $b$ -base has $t < b$ ; therefore their transverse graph coordinates occupy opposite sides of the common base plaque, with uniform nonzero first derivative after shrinking. Each half supplies a local half-neighbourhood of that plaque, and their union supplies a full neighbourhood; since the leafwise identification $h$ pairs the base inverse branches, this is precisely a neighbourhood of the seam point in the quotient $X$ .

2.1F2step 1.1

Away from the seams, $G$ is an ordinary local diffeomorphism by its two leafwise directions and the transverse time direction, so every interior point of $X$ has locally open image. Because $X$ is compact and $M$ is Hausdorff, the image $f (X)$ is closed. If $z \in f (X)$ does not lie in $f (\partial X)$ , every preimage of $z$ is an interior point and any one of them gives a neighbourhood of $z$ contained in $f (X)$ , so $z$ is not a boundary point of $f (X)$ ; hence $\partial f (X) \subseteq f (\partial X)$ .

3.1F1F2F3step 2.1∎

This proves exactly the seam openness and the image-boundary containment, allowing multiple image sheets and not substituting an immersion for an embedding. On the unglued $b$ -base annulus $D ∖ U$ the available cylinder side is $t < b$ , which is the positive transverse side when $G_{t}$ is negatively transverse, so at each regular annulus point of the unglued part the positive transverse direction points inward to the locally occupied image side; the lateral fence and possible multiple boundary sheets still require a separate no-exit argument, so no further conclusion is asserted here. The construction uses finitely many charts and inverse branches, hence only the standing countable choice from [F3].

Depends on

Used by

Dependency tree · two levels

37 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 (complete English translation) (standard reference, not scraped)