TRANSVERSE TORI IN ENGEL MANIFOLDS ROBERT E. GOMPF Abstract. We show that tori in Engel 4-manifolds behave analogously to knots in contact 3-manifolds

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TRANSVERSE TORI IN ENGEL MANIFOLDS

ROBERT E. GOMPF

Abstract. We show that tori in Engel 4-manifolds behave analogously to knots in contact 3-manifolds:

Every torus with trivial normal bundle is isotopic to inﬁnitely many distinct transverse tori, distinguished

locally (and globally in the nullhomologous case) by their formal invariants. (Few examples of transverse

tori were previously known.) We classify the formal invariants, which are richer than for transverse knots,

and show that these can all be realized by any torus in an overtwisted Engel manifold, up to homotopy

through Engel structures. Fixing Engel structures not known to be overtwisted, we explore the range of

the primary invariants of given tori. A sample application is that many Engel manifolds admit inﬁnitely

many transverse homotopy classes of unknotted transverse tori such that each class contains inﬁnitely

many transverse isotopy classes.

1. Introduction

Engel structures, which only exist on 4-manifolds, are poorly understood cousins of contact structures

in odd dimensions. The latter have been extensively studied in recent decades, ﬁrst in dimension 3 and

then in higher dimensions, and have been shown to be intimately related to the topology of the under-

lying manifolds. For example, contact topology was instrumental in proving the notorious Property P

Conjecture for 3-manifolds [KM]. One of the main tools for studying contact 3-manifolds is the notion

of a transverse knot. For example, these can be used to distinguish the crucial tight contact structures

from the less interesting overtwisted structures. The most fundamental open problem in Engel topology

is whether analogous tight Engel structures exist. If so, it seems reasonable to expect that they should

become a powerful tool for studying 4-manifolds. With this background in 2017, an AIM conference on

Engel structures was held, at which Eliashberg asked, in light of transverse knots in contact 3-manifolds,

what could be said about transverse tori in Engel manifolds. The present article shows that transverse

tori behave analogously to transverse knots, although the Engel case is richer in some aspects than the

contact case.

Engel structures naturally emerge from Cartan’s study of k-plane ﬁelds in n-manifolds [C]. A modern

exposition appears in [Pr]; we review the most relevant parts in Section 2. Cartan classiﬁed all topologically

stable subsets of the space of such distributions. By deﬁnition, these are open subsets consisting of

distributions without local invariants. That is, they are all described by the same local model at every

point. By a dimension count, there cannot be enough diﬀeomorphisms locally to generate an open set

in the space of distributions unless kor n−kis small. The most obvious case, k= 1, is the line ﬁelds

on any manifold. When n−k= 1, the topologically stable hyperplane ﬁelds are the contact structures

(nodd) and even-contact structures (neven). The only remaining case, k=n−k= 2, is the Engel

structures. These structures are all characterized as being “maximally nonintegrable” in a sense discussed

in Section 2.

Much of the power of contact topology comes from the tight/overtwisted dichotomy. Overtwisted

contact structures satisfy the h-Principle (Homotopy Principle) of Eliashberg and Gromov [EM], [Gr].

That is, there is a unique homotopy class of overtwisted contact structures within each homotopy class

of hyperplane ﬁelds endowed with suitable auxiliary data (that is vacuous when n= 3) [E], [BEM].

Thus, their classiﬁcation is a problem in algebraic topology. In contrast, tight contact structures appear

sporadically, depending in a delicate way on the topology of the underlying manifold. Even-contact

This work was inspired by the April 2017 conference “Engel Structures” at the American Institute of Mathematics. The

author would like to thank Roger Casals, Gordana Mati´c, Emmy Murphy and the other participants for helpful discussions.

arXiv:2210.04292v1 [math.GT] 9 Oct 2022

structures are considered less interesting since McDuﬀ showed they all obey the h-Principle [McD]. That

is, there are no tight even-contact structures. There are recently developed notions of overtwisted [PV]

and loose [CPP] Engel structures, both of which satisfy the h-Principle. However, the relation between

these notions is not yet known, and the fundamental question remains of whether there are additional

homotopy classes of “tight” Engel structures.

Tight contact structures on 3-manifolds are detected by which transverse knots they contain. These

are deﬁned to be circles embedded transversely to the contact planes. Immersed transverse circles satisfy

the h-Principle, with each homotopy class containing a unique transverse knot up to homotopy through

transverse immersions. However, the h-Principle fails for (embedded) transverse knots, due to a “formal”

invariant determined by the underlying bundle theory. In fact, tight contact structures are characterized

by the failure of nullhomologous knots to transversely realize large values of this invariant (and realization

need not be unique). To similarly understand Engel structures, it is natural to look for closed surfaces

transverse to an Engel plane ﬁeld. It is not hard to see (Section 3.1) that any such surface must be

a torus with trivial normal bundle. (We assume throughout the paper that everything is orientable;

otherwise transverse Klein bottles can exist, Remark 2.8.) Transverse immersions are again classiﬁed

by the h-Principle [Gr]. However, unlike circles in 3-manifolds, immersed surfaces in 4-manifolds cannot

be perturbed to embeddings, making the existence question for transverse embeddings more diﬃcult. In

fact, few examples of transverse tori were previously known. Our ﬁrst theorem, proved in Section 3.3,

solves this problem:

Theorem 1.1. A closed surface embedded in an Engel manifold is isotopic to a transverse surface if and

only if it is a torus with trivial normal bundle. If so, the isotopy can be assumed C0-small.

This raises the dual question of enumerating transverse representatives of an isotopy class, up to

isotopy through transverse embeddings, i.e., transverse isotopy. For a contact plane ﬁeld with vanishing

Euler number, nullhomologous transverse knots can be distinguished by their self-linking number in Z,

their unique formal transverse isotopy invariant. (More subtle invariants are beyond the scope of this

paper.) Every such knot is isotopic to inﬁnitely many transverse knots, distinguished by their self-

linking numbers. For homologically essential knots, the same applies in a tubular neighborhood, but not

globally in general. (A knot with a unique transverse representative appears in Example 5.13(d), with

an application to transverse tori.) The analogous invariant has been used by Kegel [K] to distinguish

certain nullhomologous transverse tori in Engel manifolds (see Example 5.7(a)). However, transverse tori

Σ also have other formal invariants. We classify the formal invariants in Section 4 (Theorem 4.3): a

pair of classes in H1(Σ) and a pair of secondary invariants in Z. Using the h-Principle for overtwisted

Engel manifolds [PV], we show (Section 5.1) that all combinations of formal invariants can be realized by

any torus in such a manifold, up to homotopy through overtwisted Engel structures. (Unlike for contact

structures on closed manifolds, a homotopy through Engel structures need not be realized by an isotopy

of the manifold.) However, it also seems worthwhile to understand the range of the formal invariants

in a ﬁxed Engel structure, especially if it is not known to be overtwisted. We analyze the analogue

∆ν∈H1(Σ) of the self-linking number in Section 5.2, leading to a general nonuniqueness theorem:

Theorem 1.2. Let Σbe a torus with trivial normal bundle in an Engel manifold M. Then

a) After a C0-small isotopy, there is a neighborhood Uin which Σis C0-small isotopic to inﬁnitely many

transverse tori, no two of which are transversely isotopic in U.

b) If [Σ] vanishes in H2(M)then no two of these tori are transversely isotopic in M.

With embedded surfaces, we encounter a subtlety that does not arise for oriented circles: There may

be an isotopy that sends Σ onto itself in a way that is nontrivial on H1(Σ). This reparametrizes Σ so that

∆νmay appear diﬀerent, even though the image surface is unchanged. (Such behavior already occurs

for unknotted tori, Example 5.7(d).) In the above theorem, and elsewhere unless otherwise speciﬁed, we

mean that the surfaces are not transversely isotopic for any choices of parametrization. This follows by

using the divisibility of ∆ν, which is preserved by automorphisms of H1(Σ). In spite of this ambiguity,

the diﬀerence Dν(F) of two values of ∆νis still well-deﬁned and useful for a ﬁxed isotopy F, and vanishes

when the isotopy is transverse. In this sense, ∆νis a transverse isotopy invariant.

By construction, the tori arising from a given Σ in the above theorem are all transversely homotopic,

that is, homotopic through immersed transverse surfaces. In particular, ∆νis not a transverse homotopy

invariant, nor is its divisibility. This reﬂects the corresponding failure of the self-linking number of

transverse knots in contact 3-manifolds, for which homotopy implies transverse homotopy. In contrast,

the other primary invariant ∆T∈H1(Σ) is a transverse homotopy invariant. Unlike ∆ν, this is well-

deﬁned even for homologically essential transverse tori, and it has no analogue for transverse knots. We

study the range of this invariant in Section 5.3. We ﬁnd that even unknotted transverse tori (isotopic to

S1×S1in some R2×R2chart) need not be transversely homotopic:

Theorem 1.3. Every circle bundle with even Euler class over a 3-manifold has a compatible Engel

structure admitting inﬁnitely many transverse homotopy classes of unknotted transverse tori such that

each class contains inﬁnitely many transverse isotopy classes.

These Engel manifolds arise by prolongation (Section 2.3) of (typically overtwisted) contact structures cho-

sen from any homotopy class of plane ﬁelds on any 3-manifold. The theorem follows from Theorem 5.11,

which provides a plethora of knot types of tori in prolongations satisfying the analogous conclusion.

The tools developed in this paper have potential application to the problem of recognizing tight

Engel structures (if any exist). This is more subtle than the corresponding problem for closed contact

manifolds, since it is not clear (to the author) whether overtwistedness is preserved by homotopy through

Engel structures. Thus, while each component Eof the space of formal Engel structures on Mcontains

a unique component of overtwisted Engel structures [PV], this may lie in a strictly larger component of

the subset of all Engel structures in E. We would ideally like to ﬁnd other components of the latter,

but a preliminary step would be to recognize any Engel structure in Ethat is not overtwisted. In the

contact setting, this can be done via restrictions on the self-linking of transverse knots, suggesting an

analogous approach for Engel structures. While this paper realizes many pairs (∆T,∆ν) for ﬁxed isotopy

classes of tori in ﬁxed Engel structures, some gaps remain. In addition, we have no concrete examples

expoiting the secondary formal invariants. For an Engel structure in E, one could hope to ﬁnd a torus

that cannot be made transverse with certain values of the formal invariants, while these values are realized

in all overtwisted Engel structures in E. (We show that such values are always realized by some such

overtwisted structure in Section 5.1.) Observation 2.7 gives a natural nested family Mr, 0 < r ≤ ∞, of

Engel manifolds diﬀeomorphic to R4such that every Engel manifold contains all bounded regions of Mr

for all suﬃciently small r. Thus, if every Mris overtwisted then all Engel manifolds are, making these the

most likely candidates for tight Engel manifolds. This suggests the utility of studying transverse torus

theory in R4, analogously to transverse knot theory in R3. The gaps in results of this paper suggest (for

example) the following questions:

Questions 1.4. a) In Mr, can an unknotted torus be transverse with ∆T6= 0? What about knotted tori?

Do the answers depend on r? In a ﬁxed Engel manifold, does every family of isotopic tori with well-deﬁned

values of ∆ν∈H1(Σ) (Corollary 4.2 and preceding) have a class δfor which each pair (∆T,∆ν+δ)is

linearly dependent (cf. Example 5.13)?

b) Is there a pair of transverse tori in Mr(or any Engel manifold) that are not transversely isotopic but

are related by an isotopy Fwith vanishing formal invariants (or just with the same ∆Tand Dν(F)=0)?

This paper is organized as follows: After reviewing the necessary background on contact and Engel

topology in Section 2, we prove Theorem 1.1 in Section 3, making a given torus transverse by isotopy.

Section 4 deﬁnes and classiﬁes the formal invariants of transverse tori. Finally, Section 5 explores the range

of the invariants, proves Theorems 1.2 and 1.3, and explicitly computes the primary invariants in some

examples. Sections 4 and 5 can be read independently of Section 3, and proofs of the latter two theorems

and the analysis of the primary invariants (Sections 5.2 and 5.3) can be read without Sections 4.2, 4.3 and

5.1. We use the following conventions throughout the paper, except where otherwise indicated: Homology

and cohomology have integral coeﬃcients, with PD denoting Poincar´e duality or its inverse. We work in

the category of smooth, connected manifolds. Curves and surfaces are assumed to be closed (and the latter

are often tori). Other manifolds are allowed to be noncompact but (for simplicity) without boundary.

Manifolds and distributions on them are assumed to be oriented, compatibly. The exact meaning of

compatibility is only needed for some signs in Sections 5.2 and 5.3. However, for future research it seems

important to specify the meaning in a way that is optimally compatible with the standard conventions

of smooth and contact topology, so we discuss the details carefully in Section 2.4.

2. Background

This section reviews various ways of visualizing and manipulating contact 3-manifolds, Engel 4-

manifolds and their submanifolds, as well as presenting Lemma 2.3 and Addendum 2.5 for later use,

and establishing natural orientation conventions for Engel topology. To begin, we must understand the

meaning of “maximal nonintegrability” that characterizes topologically stable distributions. We consider

hyperplane ﬁelds here and Engel 2-plane ﬁelds in Section 2.3. For the former, maximal nonintegrability

means that the Lie bracket operation [u, v] on vector ﬁelds is maximally nondegenerate on the hyperplane

ﬁeld: Every (orientable) hyperplane ﬁeld in a manifold is the kernel of some 1-form α, unique up to scale.

The standard formula

dα(u, v) = uα(v)−vα(u)−α[u, v]

implies that when uand vare vector ﬁelds in ker α,dα(u, v) = −α[u, v] = α[v, u]. This interprets the

normal component of the Lie bracket on ker αas a pointwise, bilinear form. Maximal nonintegrability of

a contact or even-contact structure is then maximal nondegeneracy of dα|ker α. For contact structures,

ker αhas even dimension, so this means dα|ker αis symplectic. On a 3-manifold N, this just says dα is

never 0 on the 2-planes ξ, so it is an area form on them that we always assume is positive. Equivalently,

α∧dα is a positive volume form on N. For an even-contact structure E, the hyperplanes have odd

dimension, so maximal nondegeneracy means there is a canonical line ﬁeld Win Esuch that dα descends

to a well-deﬁned symplectic form on E/W. On a 4-manifold M, this occurs whenever dα|E is never 0. In

terms of Lie brackets, nonintegrability on 3- and 4-manifolds is given by the conditions [ξ, ξ] = T N and

[E,E] = T M, respectively. For even-contact structures of any dimension, Wis equivalently characterized

by the condition [W,E]⊂ E. Thus, the ﬂow of any vector ﬁeld in Wpreserves E, since its Lie derivative

preserves the set of vector ﬁelds in E. On any open subset of Mwhose quotient by such a ﬂow is a

manifold, the latter then canonically inherits a contact structure ξ=E/W. Similarly, any hypersurface

N⊂Mtransverse to Winherits a contact structure ξ=E ∩ T N that is invariant under such ﬂows and

locally projects to the canonical contact structure E/W.

2.1. Contact topology. We now review contact 3-manifolds and their submanifolds, deferring the fun-

damental topic of convex surfaces to Section 2.2. (See, e.g., [OS] for more details.) First we consider

knots suitably compatible with the ambient (oriented) contact plane ﬁeld ξ= ker α.Transverse knots are

everywhere transverse to ξ, whereas Legendrian knots are everywhere tangent to ξ. If Kis transverse, it

is canonically oriented by the condition α|K > 0, whereas a Legendrian Khas α|K= 0 everywhere, so

can be oriented arbitrarily. Transverse knots have a unique formal invariant. This arises from a relative

invariant associated to each regular homotopy between transverse knots (namely, the diﬀerence between

relative Euler classes of ξ|Kand the normal bundle νK pulled back over the domain I×S1). When

Kis nullhomologous and the Euler class e(ξ) vanishes, it becomes an absolute invariant, the self-linking

number l(K)∈Z. (When e(ξ)6= 0, this is still deﬁned relative to a preassigned Seifert surface Σ since

e(ξ|Σ) = 0.) The self-linking number is deﬁned to be the winding number along Kof any nowhere-zero

section of ξon N, relative to the 0-framing (which is a vector ﬁeld outward normal to any Seifert surface,

that we can assume lies in ξsince Kis transverse). Consideration of spin structures shows that this is

always odd. A similar discussion of Legendrian knots yields two formal invariants tb(K), r(K)∈Z. We

only need tb(K), which measures a vector ﬁeld transverse to ξalong Krelative to the 0-framing. We

can now characterize tight contact structures in four ways: There is no transverse unknot with l(K)≥0

or Legendrian unknot with tb(K)≥0, and every nullhomologous knot has an upper bound on tb of

Legendrian representatives, or (if e(ξ) = 0 or for a ﬁxed Seifert surface) on lof transverse representatives.

To construct local models of subsets of contact manifolds, consider the standard contact structure

on R3, which we usually describe as the kernel of α=dz +xdy. This is the unique tight contact

Figure 1. Stabilizing a transverse knot by a C0-small isotopy.

structure on R3up to contactomorphism (diﬀeomorphism preserving the contact plane ﬁeld), although

we sometimes describe it using other contactomorphic plane ﬁelds. Uniqueness guarantees that every

point of a contact 3-manifold has a neighborhood contactomorphic to the standard R3. We will introduce

various other local models as needed. For example, every Legendrian knot in a contact 3-manifold has

a neighborhood pairwise contactomorphic to a neighborhood of the y-axis in (R3, dz +xdy) mod unit

y-translation. Similarly, every transverse knot has a neighborhood contactomorphic to a neighborhood

of the z-axis mod unit z-translation. In the latter case, it is a bit more natural to use the cylindrically

symmetric contact form given by α0=dz +1

2(xdy −ydx) = dz +1

2r2dθ in cylindrical coordinates. This

is related to αby the contactomorphism ϕ(x, y, z) = (x, y, z +1

2xy) with ϕ∗(α0) = α.

Knots in (R3, dz +xdy) can be described by projections. The front projection (x, y, z)7→ (y, z) sends

each contact plane to a line, whose slope dz

dy =−xrecovers the deleted coordinate. These lines realize all

nonvertical slopes and are co-oriented upward. We can now represent a knot by its image in the projection,

together with lines whose slopes recover xas in Figure 1. Then a transverse knot projects to an immersion

that is everywhere positively transverse to the lines as in the ﬁgure. (Note that a positive crossing with

both strands oriented downward cannot occur.) The self-linking number is then the winding number of

the blackboard framing given by a vector ﬁeld in ξparallel to the x-axis, relative to the 0-framing; this is

just the signed number of crossings in the diagram. A generic knot has ﬁnitely many tangencies to ξthat

become tangencies to lines in the projection. The image of a Legendrian knot is everywhere tangent to

the lines, which then need not be drawn. Thus, we recover the x-coordinate from its slope everywhere.

Since the slope cannot be vertical, the projection must have cusps at which the knot is parallel to the

x-axis. The formal invariants can be read from the diagram by suitably counting crossings and cusps.

A Legendrian knot can also be described by its Lagrangian projection (x, y, z)7→ (x, y). We recover the

z-coordinate up to a constant as ∆z=−Rxdy. The integral can be interpreted as a signed area by

Green’s Theorem. This must vanish when we traverse the entire image of a closed Legendrian curve in

R3. Similarly, two Legendrian arcs Cand C0with the same initial point will have the same endpoint if

and only if these endpoints have the same Lagrangian projection and the enclosed signed area vanishes.

There are several natural operations on Legendrian and transverse knots. Both kinds of knots can be

stabilized, lowering tb by 1 or lby 2, respectively. Figure 1 shows the front projection of this procedure

for a transverse knot in the standard R3, and is a local model for the general case. The altered curve

can be kept transverse by suitably controlling the x-coordinate (as measured by ξin the ﬁgure). By

keeping the new loops narrow and changes in xsmall, we can arrange the stabilization to result from

either an arbitrarily C0-small isotopy, or a C0-small transverse homotopy. The homotopy carries along

the blackboard framing on ξ, but the isotopy twists the Seifert surface, so the self-linking drops by 2 (as

also seen by counting crossings with sign). Another natural operation is the transverse pushoﬀ changing

an oriented Legendrian knot Kto a transverse knot τK. More precisely, there is an embedding of R×S1

such that {t} × S1maps onto Kwhen t= 0 and is transverse otherwise, with orientation depending on

the sign of t(so each orientation on Kis realized as a transverse pushoﬀ). The annulus is apparent in

the image of the xy-plane in the local model of Kfrom two paragraphs previously. Alternatively, this

operation can be derived using the contact condition and Lie derivative. We will adapt the latter method

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TRANSVERSETORIINENGELMANIFOLDSROBERTE.GOMPFAbstract.WeshowthattoriinEngel4-manifoldsbehaveanalogouslytoknotsincontact3-manifolds:Everytoruswithtrivialnormalbundleisisotopictoinnitelymanydistincttransversetori,distinguishedlocally(andgloballyinthenullhomologouscase)bytheirformalinvariants.(Fewexampl...

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TRANSVERSE TORI IN ENGEL MANIFOLDS ROBERT E. GOMPF Abstract. We show that tori in Engel 4-manifolds behave analogously to knots in contact 3-manifolds

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