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An alternative proof of the existence of totally real embeddingsof 3‐manifolds into \mathb {C}^{3}

Naohiko Kasuya

Department of Mathematics, Kyoto Sangyo University

1Introduction

Let M^{n} be a closed, connected, orientable n ‐manifold and f:M^{n}arrow \mathbb{C}^{n} be an immersion.

A point p\in M^{n} is said to be a complex tangent if df_{p}(T_{p}M^{n}) contains a complex line.
By Thom’s transversality theorem, the set of complex tangents of a generic immersion
f:M^{n}arrow \mathbb{C}^{n} is empty or forms a closed (n-2)‐dimensional submanifold. Elgindi initiatedto study the problem of determining the isotopy classes of knots in S^{3} which can be realized

as the set of complex tangents of an embedding S^{3}arrow \mathbb{C}^{3}[2,3,4] . In [11] the aurhor andTakase showed that any link L in a closed oriented 3‐manifold M^{3} can be realized as theset of complex tangents of an embedding M^{3}arrow \mathbb{C}^{3} if and only if the homology class [L]is trivial in H_{1}(M^{3};\mathbb{Z}) .
An immersion is said to be totally real if it has no complex tangent, and when theimmersion is embedding, it is called a totally real embedding. For totally real embeddings,

Gromov [8] and Forstnerič [6] proved the following theorem. This is called the h ‐principle
for totally real embeddings.

Theorem 1 (Gromov [8], Forstnerič [6]). Let M^{n} be a closed orientable n ‐manifold withn\geq 3 . Then, M^{n} admits a totally real embedding into \mathbb{C}^{n} if and only if it admits a totallyreal immersion into \mathbb{C}^{n} which is regularly homotopic to an embedding.

As a consequence of this theorem, the following is easily shown.
Corollary 2. Any closed orientable 3‐manifold admits a totally real embedding into \mathbb{C}^{3}.
Since the proof relies on theh ‐principle,

however, almost nothing can be analyzed

about the obtained totally real embedding. On the other hand, some explicit examples

of totally real embedings are known. Ahern and Rudin [1] explicitly constructed a totallyreal embedding of the 3‐sphere into \mathb {C}^{3} . In this article, using Ahern‐Rudin’s example, wegive a new proof of Corollary 2. The argument of the proof was inspired by the work of69
Etnyre and Furukawa [5]. They defined the notion of braided embeddings and used it toprove the existence of contact embeddings into the standard contact 5‐sphere for somecontact 3‐manifolds. We show that braided embeddings are also useful for constructing
totally real embeddings.

2Preliminaries

In this section, we introduce Ahern‐Rudin’s example and the notion of braided embed‐
dings.

In [7] Gromov stated that there exist totally real embeddings of the 3‐sphere into \mathb {C}^{3},but he did not give the proof there. In order to prove it, Ahern and Rudin [1] constructed
the following example.

Example 3 (Ahern‐Rudin [1]). Let P(z, w)=\overline{z}\overline{w}(|w|^{2}+i|z|^{2}) . We consider the 3‐
sphere as the unit shpere S^{3}=\{(z, w)||z|^{2}+|w|^{2}=1\}\subset \mathbb{C}^{2} . Then, the embedding
F:S^{3}arrow \mathbb{C}^{3} defined byF(z, w)=(z, w, P(z, w))
is a totally real embedding.
Next, we explain the definition of braided embeddings. First, we recall branched cover‐ings.

Definition 4. Let M^{n} and Y^{n} be n ‐manifolds. A d ‐fold branched covering is a smooth,proper map p :M^{n}arrow Y^{n} with critical set B\subset Y^{n}called the branch locus, such thatM^{n}-p^{-1}(B) is a covering map of degree d , and for each x\in p^{-1}(B) there
are local coordinates near x and p(x) such that p is given by (q, z)\mapsto(q, z^{m}) for somep restrictedm\in \mathbb{Z}_{>0} , where q is a coordinate on D^{n-2} and z is a coordinate on the unit disk in \mathbb{C}.

The integer m is called the branching index of p at x. A d‐fold branched covering is called
simple if the pre‐image of any point in Y^{n} has either d ord-1 points.

Etnyre and Furukawa [5] defined the following notion.
Definition 5 (Etnyre‐Furukawa [5]). Let M^{n} and Y^{n} be n ‐manifolds. An embeddinge:M^{n}arrow Y^{n}\cross D^{2}is called a braid about Y^{n} if \pi\circ e : M^{n}arrow Y^{n} is a branched covering, where \pi : Y^{n}\cross D^{2}arrowY^{n}is the first projection. If Y^{n} is embedded in a(n+2) ‐manifold W^{n+2} with trivial

normal bundle, then M^{n} is also embedded in W^{n+2} . This embedding of M^{n} into W^{n+2}70
is called a braided embedding. Moreover, a branched covering p:M^{n}arrow Y^{n} is said to bebraided about Y^{n} if there exists a function

f : M^{n}arrow D^{2} such thatM^{n}arrow Y^{n}\cross D^{2} : x\mapsto(p(x), f(x))

e :is an embedding.

For braided embeddings of 3‐manifolds, a theorem due to Hilden, Lozano and Mon‐tesinos [10] is known. Using the terminology of [5], their theorem can be stated as follows.
Theorem 6 (Hilden‐Lozano‐Montesinos [10]). Every closed oriented 3‐manifold M^{3} canbe braided about the 3‐sphere where the corresponding branched covering is a simple

3‐fold branched covering.

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An alternative proof of Corollary 2Combining Example 2 with Theorem 6, we can give a very simple proof of Corollary 2.

Proof. In Example 2, F also defines an embedding of \mathb {C}^{2} into \mathb {C}^{3} . Since the normal bundleof the embedding F is trivial, we obtain an embedding \overline{F} of a tubular neibourhood \mathbb{C}^{2}\cross D^{2}
into \mathb {C}^{3} . We also describe the restricted embedding S^{3}\cross D^{2}arrow \mathbb{C}^{3} by the same symbol\overline{F} . Then, of course, \overline{F}(S^{3}\cross\{(0,0)\})=F(S^{3}) is nothing but Ahern‐Rudin’s example. ByTheorem 6, for any closed orientable 3‐manifold M^{3} , there is a function f : M^{3}arrow D^{2}such thate :M^{3}arrow S^{3}\cross D^{2} : x\mapsto(p(x), f(x))

is an embedding, where p : M^{3}arrow S^{3} is a simple 3‐fold branched covering. Since thetotally reality is an open condition, for a sufficiently small positive number \epsilon , the tangent

space of the image of the embedding
e_{\epsilon}:M^{3}arrow S^{3}\cross D^{2} : x\mapsto(p(x), \epsilon f(x))

is close enough to that of S^{3}\cross\{(0,0)\} in the sense of C^{\infty} ‐topology, so that the composition

with the embedding \overline{F}:S^{3}\cross D^{2}arrow \mathbb{C}^{3} is a totally real embedding. Thus, we obtained atotally real embedding \overline{F}\circ e_{\epsilon} : M^{3}arrow \mathbb{C}^{3}.

\squareAlthough the above proof is not by an explicit construction in the sense that the function
f is not explicitly given, further analysis of the obtained totally real embedding can be
expected because we avoided using theh ‐principle.

For example, it might be easy totake a Seifert surface of the totally real submanifold, since the corresponding branchedcovering carries informations of the totally real embedding. However, there is a problem.
The embedding \overline{F}\circ e_{\epsilon} : M^{3}arrow \mathbb{R}^{6} arises from an embedding of M^{3} into \mathbb{R}^{5} . Hence,

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intersting examples like Haefliger knots [9] never appears. In order to realize Haefliger
knots as totally real submanifolds explicitly, we need to study braided immersions or the3‐codimensional version of braided embeddings of 3‐manifolds.
Problem 7. Can a Haefliger knot be realized as a 3‐codimensional braided embeddingof the 3‐sphre?

The author suspect that Takase’s works on Haefliger knots [12, 13] are the keys toapproaching this problem. Anyway this is a future problem.
References[1] Patrick Ahern and Walter Rudin, Totally real embeddings of S^{3} in C^{3} , Proc. Amer.
Math. Soc. 94 (1985) 460‐462.
[2] Ali M. Elgindi. On the topological structure of complex tangencies to embeddings ofS^{3} into C^{3} , New York J. Math. 18 (2012) 295‐313.

[3] Ali M. Elgindi, A topological obstruction to the removal of a degenerate com‐plex tangent and some related homotopy and homology groups, Internat. J. Math.

26(5):1550025, 16, 2015.
[4] Ali M. Elgindi, Totally real perturbations and non‐degenerate embeddings of S^{3} , NewYork J. Math. 21 (2015) 1283‐1293.
[5] John Etnyre and Ryo Furukawa, Braided embeddings of contact 3‐manifolds in thestandard contact 5‐sphere, Journal of Topology 10 (2017), 412‐446.

[6] Franc Forstnerič, On totally real embeddings into C^{n} , Exposition. Math. 4 (1986),
no. 3, 243‐255.

[7] M. Gromov, Convex integration of differential relations, Math. USSR‐Izv. 7 (1973),329‐343.

[8] Mikhael Gromov, Partial differential relations, Ergebnisse der Mathematik und ihrer
Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 9, Springer‐Verlag, Berlin, 1986.

[9] A. Haefliger, Knotted (4k -1) ‐spheres in 6k ‐space, Ann. of Math. 75 (1962) 452‐466.
[10] Hugh M. Hilden, María Teresa Lozano, and José María Montesinos, All three‐manifolds are pullbacks of a branched covering S^{3} to S^{3} , Trans. Amer. Math. Soc.

279 (1983) 729‐735.

72[11] N. Kasuya and M. Takase, Knots and links of complex tangents, Trans. Amer. Math.
Soc. 370 (2018) 2023‐2038.
[12] M. Takase, A geometric formula for Haefliger knots, Topology 43 (2004) 1425‐1447.

[13] M. Takase, The Hopf invariant of a Haefliger knot, Mathematische Zeitschrift 256(2007) 35‐44.
Department of Mathematics

Faculty of Science, Kyoto Sangyo University
Motoyama, Kamigamo, Kita‐ku

Kyoto 603‐8555 JAPANE‐mail address: nkasuya@cc.kyoto‐su.ac.jp