Classification of tangent and transverse knots in bracket-generating distributions_2

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CLASSIFICATION OF TANGENT AND TRANSVERSE KNOTS IN

BRACKET-GENERATING DISTRIBUTIONS

JAVIER MART´

INEZ-AGUINAGA AND ´

ALVARO DEL PINO

Abstract. Consider a manifold, of dimension greater than 3, equipped with a bracket-generating

distribution. In this article we prove complete h-principles for embedded regular horizontal curves

and for embedded transverse curves. These results contrast with the 3-dimensional contact case,

where the full h-principle for transverse/legendrian knots is known not to hold.

We also prove analogous statements for immersions, with no assumptions on the ambient dimen-

sion.

Contents

1. Introduction 1

2. Preliminaries on distributions 5

3. ε-horizontality and ε-transversality 7

4. Graphical models 11

5. Microﬂexibility of curves 15

6. Tangles 21

7. Controllers 34

8. h-Principles for horizontal curves 37

9. h-Principle for transverse embeddings 44

10. Appendix: Technical lemmas on commutators 48

References 50

1. Introduction

1.1. Setup. Aq-distribution on a smooth manifold Mis a smooth section Dof the Grassmann bundle

of q-planes. There is a control-theoretic motivation for considering such objects: we may think of M

as conﬁguration space and of Das the admissible directions of motion. Then, a natural question is

whether any two points in Mcan be connected by a horizontal path, i.e. a path whose velocity vectors

take values in D. A suﬃcient condition is given by a classic theorem of Chow [12]: any two points in

Mcan be connected if Dis bracket-generating. Being bracket-generating means that any vector in

T M can be written as a linear combination of Lie brackets involving sections of D. That is, Chow’s

theorem is an inﬁnitesimal to global statement.

Even though classic proofs of Chow’s theorem produce horizontal paths that are piecewise smooth,

C∞-paths can be constructed by suitable smoothing, see [27, Subsection 1.2.B]. It follows that every

homotopy class of loops on Mcan be represented by a smooth horizontal loop. That is, the inclusion

ι:L(M, D)−→ L(M)

2010 Mathematics Subject Classiﬁcation. Primary: 53D10. Secondary: 53D15, 57R17.

arXiv:2210.00582v1 [math.DG] 2 Oct 2022

2 JAVIER MART´

INEZ-AGUINAGA AND ´

ALVARO DEL PINO

is a π0-surjection. Here L(M) is the (unbased) loop space of M(endowed with the C∞-topology)

and L(M, D) is the subspace of horizontal loops. More recently, Z. Ge [23] proved that the analogous

inclusion for H1-loops is a weak homotopy equivalence; see also [9].

In this paper we consider a variation on this theme, proving classiﬁcation statements for spaces of hor-

izontal embeddings. Our theorems relate these spaces to their formal counterparts (roughly speaking,

spaces of smooth embeddings plus some additional homotopical data). Taking care of the embedding

condition is rather delicate (as is often the case for h-principles of this type) and much of the paper is

dedicated to handling it. Analogous classiﬁcation statements hold for horizontal immersions, with sim-

pler proofs. We also deduce that the map ιabove is a weak homotopy equivalence (i.e. the smooth

analogue of Ge’s theorem). Lastly, our techniques translate to the setting of embedded/immersed

transverse curves, yielding similar classiﬁcation results.

We now state our theorems. We work under the following assumption:

Assumption 1.1. All the bracket-generating distributions we consider in this paper are of constant

growth (i.e. the growth vector does not depend on the point). See Subsection 2.1.2.

1.2. Immersed horizontal curves. Let us write Imm(M, D)⊂ L(M, D) for the subspace of im-

mersed horizontal loops. In order to study it, we introduce the so-called scanning map:

Imm(M, D)−→ Immf(M, D),

taking values in the space of formal horizontal immersions

Immf(M, D) := {(γ, F )|γ∈ L(M), F ∈Mon(TS1, γ∗D)}.

The question to be addressed is whether the scanning map is a weak homotopy equivalence. The

answer is positive if Dis a contact structure [18, Section 14.1] but, for other distributions, the answer

may be negative due to the presence of rigid curves [6].

A horizontal curve is rigid if possesses no C∞-deformations relative to its endpoints, up to reparametri-

sation. These curves are isolated and conform exceptional components within the space of all hori-

zontal maps with given boundary conditions. Rigid loops also exist. Because of this, the inclusion

Imm(M, D)→Immf(M, D) can fail to be bijective at the level of connected components; see [36,

Remark 23]. Being rigid is the most extreme case of being singular. This means that the endpoint

map of the curve is not submersive, so the curve has fewer deformations than expected; see Subsection

2.2.

The subspaces of rigid and singular curves have a geometric and not a topological nature. By this,

we mean that small perturbations of Dcan radically change their homotopy type; see [32] or [36,

Theorem 27]. This motivates us to discard singular curves and focus on Immr(M, D), the subspace of

regular horizontal immersions. In doing so, the subspace that we discard is not too large: Germs of

singular horizontal curves were shown to form a subset of inﬁnite codimension among all horizontal

germs, ﬁrst in the analytic case [35] and then in general [8]. Earlier, it had already been observed [29,

Corollary 7] that regular (i.e. non-singular) germs are C∞–generic.

Our ﬁrst result reads:

Theorem 1.2. Let (M, D)be a manifold endowed with a bracket-generating distribution. Then, the

following inclusion is a weak homotopy equivalence:

Immr(M, D)−→ Immf(M, D).

Apart from the aforementioned contact case, in which there are no singular curves, this was already

known in the Engel case [36].

Corollary 1.3. Let D0and D1be bracket-generating distributions on a manifold M, homotopic as

subbundles of T M . Then, the spaces Immr(M, D0)and Immr(M, D1)are weakly homotopy equivalent.

This follows immediately from Theorem 1.2 and the analogous fact about Immf(M, D0) and Immf(M, D1).

It follows that all the data about Dencoded in Immr(M, D) is purely formal.

CLASSIFICATION OF TANGENT AND TRANSVERSE KNOTS IN BRACKET-GENERATING DISTRIBUTIONS 3

1.3. Embedded horizontal curves. We now consider the subspace of embedded horizontal loops

Emb(M, D)⊂Imm(M, D), together with its scanning map

Emb(M, D)−→ Embf(M, D),

into the space of formal horizontal embeddings:

Embf(M, D) := γ, (Fs)s∈[0,1]:γ∈Emb(M), Fs∈MonS1(TS1, γ∗T M),

F0=γ0, F1∈γ∗D},

i.e. the homotopy pullback of Emb(M) and Immf(M, D) mapping into Immf(M). Reasoning as

above leads us to introduce Embr(M, D), the subspace of regular horizontal embeddings. Our second

(and main) result reads:

Theorem 1.4. Let (M, D)be a bracket-generating distribution with dim(M)≥4. Then, the following

inclusion is a weak homotopy equivalence:

Embr(M, D)−→ Embf(MD).

Note that the dimensional assumption is sharp, since the result is known to be false in 3-dimensional

Contact Topology [4].

Theorem 1.4 was already known in the Engel case [15] and in the higher-dimensional contact setting

[18, p. 128]. Our arguments diﬀer considerably from both. The proof in [18] is contact-theoretical

in nature, relying on isocontact immersions. The one in [15] uses the so-called Geiges projection,

which is particular to the Engel case. The methods in the present paper use instead local charts in

which the distribution can be understood as a connection; see Subsection 4. This is reminiscent of

the Lagrangian projection in Contact Topology and closely related to methods used in the Geometric

Control Theory [27,32] (with the added diﬃculty of tracking the embedding condition).

Much like earlier:

Corollary 1.5. Fix a manifold Mwith dim(M)≥4. Let D0and D1be bracket-generating distribu-

tions on M, homotopic as subbundles of T M. Then, the spaces Embr(M, D0)and Embr(M, D1)are

weakly homotopy equivalent.

1.4. Horizontal loops. Now we go back to the problem we started with:

Theorem 1.6. Let (M, D)be a manifold endowed with a bracket-generating distribution. Then, the

following inclusion is a weak homotopy equivalence:

L(M, D)−→ L(M).

This also holds for the (based) loop space Ωp(M) and its subspace of horizontal loops Ωp(M, D),

for all p∈M. Observe that the statement uses no regularity assumptions. The reason is that

singularity issues can be bypassed thanks to what we call the stopping-trick (namely, one can slow

the parametrisation of a horizontal curve down to zero locally in order to guarantee that enough

compactly-supported variations exist). See Subsection 8.5.

1.5. Immersed transverse curves. The other geometrically interesting notion for curves in bracket–

generating distributions is that of transversality. We deﬁne ImmT(M, D) to be the space of immersed

loops that are everywhere transverse to D. Like in the horizontal setting, one can introduce formal

transverse immersions

Immf

T(M, D) = {(γ, F ) : γ∈ L(M), F ∈MonS1(TS1, γ∗(T M/SD))},

and see that there is a scanning map

ImmT(M, D)−→ Immf

T(M, D).

Being transverse is an open condition and therefore rigidity/singularity is not a phenomenon we

encounter. We prove:

4 JAVIER MART´

INEZ-AGUINAGA AND ´

ALVARO DEL PINO

Theorem 1.7. Let (M, D)be a manifold endowed with a bracket-generating distribution. Then the

inclusion

ImmT(M, D)−→ Immf

T(M, D)

is a weak homotopy equivalence.

This result is not new. The h-principle for smooth immersions (of any dimension!) transverse to

analytic bracket-generating distributions was proven in [35]. The analyticity assumption was later

dropped by A. Bhowmick in [8], using Nash-Moser methods. Both articles rely on an argument due to

Gromov relating the ﬂexibility of transverse maps to the microﬂexibility of (micro)regular horizontal

curves. The approach in this paper is independent.

Once again, a corollary is that the weak homotopy type of ImmT(M, D) depends on Donly formally.

1.6. Embedded transverse curves. Lastly, we address embedded transverse loops EmbT(M, D)

and their scanning map into the analogous formal space:

Embf

T(M, D) = γ, (Fs)s∈[0,1]:γ∈Emb(M), Fs∈MonS1(TS1, γ∗T M),

F0=γ0, F1:TS1→γ∗T M →γ∗(T M/D) is injective .

Our fourth result reads:

Theorem 1.8. Let (M, D)be a bracket-generating distribution with dim(M)≥4. Then the inclusion

EmbT(M, D)−→ EmbTf(M, D)

is a weak homotopy equivalence. In particular, EmbT(M, D)depends only on the formal class of D.

The dimension condition is sharp, since transverse embeddings into 3-dimensional contact manifolds

do not satisfy a complete h-principle. Indeed, there are examples of transverse knots that have

the same formal invariants but are not transversely isotopic [7]. Furthermore, Theorem 1.8 is only

interesting in corank 1. Indeed, it is a classic result [18, 4.6.2] that closed n-dimensional submanifolds

transverse to corank kdistributions abide by all forms of the h−principle if k > n.

1.7. Structure of the paper. In Section 2we recall some standard deﬁnitions from the theory

of tangent distributions. Basics of h-principle and some preliminary results, using the theory of

ε-horizontal embeddings, are presented in Section 3.

In Section 4we introduce the notion of graphical model. These are local descriptions in which the

distribution is seen as a connection. Many of our arguments take place in such a local setting. Section

5contains a series of technical lemmas (that roughly speaking correspond to the “reduction step” in

our h-principles) about manipulating families of curves.

Sections 6and 7contain the main technical ingredients behind the proof, the notions of tangle and

controller. These are models for horizontal curves (or rather, models for their projections to the base

of a graphical model) meant to be used to produce a displacement transverse to the distribution.

They play a role analogous to the stabilisation in Contact Topology, except for the fact that they can

be introduced through homotopies of embedded horizontal curves. The existence of such a homotopy

uses strongly the fact that the ambient dimension is at least 4 (and it is still rather technical to

implement).

The h-principles for horizontal curves are proven in Section 8. The h-principles for transverse curves

in Section 9. Along the way we state and prove the appropriate relative versions. We will put all

our emphasis on the embedding cases; the other statements (immersions and simply smooth curves)

follow from the same arguments with considerable simpliﬁcations.

Appendix 10 contains various technical results on commutators of vector ﬁelds. These are used often

throughout the paper.

Acknowledgments: The authors are thankful to E. Fern´andez, M. Crainic, F. Presas, and L.

Toussaint for their interest in this project. During the development of this work the ﬁrst author was

supported by the “Programa Predoctoral de Formaci´on de Personal Investigador No Doctor” scheme

CLASSIFICATION OF TANGENT AND TRANSVERSE KNOTS IN BRACKET-GENERATING DISTRIBUTIONS 5

funded by the Basque department of education (“Departamento de Educaci´on del Gobierno Vasco”).

The second author was funded by the NWO grant 016.Veni.192.013; this grant also funded the visits

of the ﬁrst author to Utrecht.

2. Preliminaries on distributions

In this section we recall some of the basic theory of distributions, including the notions of singularity

and rigidity for horizontal curves (Subsection 2.2). For further details we refer the reader to [28,32].

2.1. Diﬀerential systems. The following deﬁnition generalises the notion of distribution:

Deﬁnition 2.1. Let Mbe a smooth manifold. A diﬀerential system Dis a C∞-submodule of the

space of smooth vector ﬁelds.

Given a smooth distribution on M, we can construct a diﬀerential system by taking its smooth sections.

Conversely, a diﬀerential system Darises from a distribution if the dimension of its pointwise span

D(p)⊂TpMis independent of p∈M. In this manner, we think of diﬀerential systems as singular

distributions; we will often abuse notation and use Dto denote both the distribution and its sections.

Remark 2.2. When Mis not compact, it is convenient to impose that Dsatisﬁes the sheaf condition.

The reason is that there may be diﬀerential systems that only diﬀer from one another due to their

behaviour at inﬁnity; imposing the sheaf condition removes this redundancy. These subtleties will not

be relevant for us.

2.1.1. Lie ﬂag. Let us introduce some terminology. We say that the string a, depending on the

variable a, is a formal bracket expression of length 1. Similarly, we say that the string [a1, a2],

depending on the variables a1and a2, is a formal bracket expression of length 2. Inductively, we

deﬁne a formal bracket expression of length nto be a string of the form [A(a1,··· , aj), B(aj+1, an)]

with 0 < j < n and Aand Bformal bracket expressions of lengths jand n−j, respectively.

Given a diﬀerential system D, we deﬁne its Lie ﬂag as the sequence of diﬀerential systems

D1⊂ D2⊂ D3⊂ ···

in which Diis the C∞-span of vector ﬁelds of the form A(v1,··· , vj), j≤i, where the vkare vector

ﬁelds in Dand Ais a formal bracket expression of length j. As such, D1=D.

2.1.2. Growth vector. Given a point p∈M, one can use the Lie ﬂag to produce a ﬂag of vector spaces:

D1(p)⊂ D2(p)⊂ D3(p)⊂ ···

Here Di(p) denotes the span of Diat p. This yields a non-decreasing sequence of integers

(dim(D1(p)),dim(D2(p)),dim(D3(p)),···)

which in general depends on p. This sequence is called the growth vector of Dat p.

If the growth vector does not depend on the point, we will say that the diﬀerential system Dis of

constant growth. If this is the case, all the diﬀerential systems in the Lie ﬂag arise as spaces of

sections of distributions. Some examples of distributions of constant growth are (regular) foliations,

contact structures, and Engel structures.

The following notion is central to us:

Deﬁnition 2.3. A diﬀerential system (M, D)is bracket-generating if, for every p∈Mand every

v∈TpM, there is an integer msuch that v∈ Dm(p). This integer is called the step.

As stated in Assumption 1.1: we henceforth focus on bracket-generating distributions of constant

growth.

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CLASSIFICATIONOFTANGENTANDTRANSVERSEKNOTSINBRACKET-GENERATINGDISTRIBUTIONSJAVIERMARTINEZ-AGUINAGAANDALVARODELPINOAbstract.Consideramanifold,ofdimensiongreaterthan3,equippedwithabracket-generatingdistribution.Inthisarticleweprovecompleteh-principlesforembeddedregularhorizontalcurvesandforembeddedtr...

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