Problems on Low-dimensional Topology, 2018
Edited by T. Ohtsuki1

This is a list of open problems on low-dimensional topology with expositions of
their history, background, significance, or importance. This list was made by editing
manuscripts written by contributors of open problems to the problem session of theconference “Intelligence of Low-dimensional Topology” held at Research Institutefor Mathematical Sciences, Kyoto University in May 30 – June 1, 2018.

Contents
1 Computational complexity of the colored Jones polynomial at a2
root of unity
2 The minimal coloring number of Z-colorable links

3

3 Triangulations and the 3D index of cusped 3-manifolds

4

4 Destabilized Heegaard surfaces of 3-manifolds

5

5 Simplified trisections of 4-manifolds

6

6 Virtual embeddings between mapping class groups of surfaces

9

7 Totally real immersions and embeddings of n-manifolds into Cn

11

1 Research Institute for Mathematical Sciences, Kyoto University, Sakyo-ku, Kyoto, 606-8502, JAPAN

Email: tomotada@kurims.kyoto-u.ac.jp
The editor is partially supported by JSPS KAKENHI Grant Numbers JP16H02145 and JP16K13754.

11Computational complexity of the colored Jones polynomial at a root of unity(Oliver Dasbach)
The volume conjecture of Kashaev and Murakami & Murakami and its generalizations
the growth of evaluations of the colored Jones polynomial√ connect)(
JN L; e2π −1/N of a link L to geometric information of its link complement. For
J2 (L; q), i.e. the classical Jones polynomial, it is known by a result of Jaeger, Vertigan and Welsh [32, 61] that evaluations at all but eight points (2nd, 3rd, 4th, 6throots of unity) are #P-hard. At those eight points polynomial time algorithms are
known (see the second remark below), in particular for J2 (L; −1).
Question
1.1(O.) Dasbach). What can one say about the computational complexity√(of JN L; e2π −1/N for N = 5 and N ≥ 7?
Remark. It is known (see e.g. [50]) that the colored Jones polynomial JN (K; q) ofa knot K can be presented by a linear sum of the Jones polynomial J2 (K (n) ; q) of
K (n) (for n = 1, 2, · · · , N − 1), where K (n) denotes the union of n parallel copies of
K. Similarly, JN (L; q) of a link L can be presented by a linear sum of the Jones
polynomial of links obtained from L by replacing each component with parallel
copies of it. Further, it is known (see the remark below) that the Jones polynomial
can be calculated in polynomial√ time() at 2nd, 3rd, 4th, 6th roots of unity. Hence,
when N = 2, 3, 4, 6, JN L; e2π −1/N can be calculated in polynomial time.
Remark. At a 2nd, 3rd, 4th, 6th root of unity, the Jones polynomial of a link L canbe calculated in polynomial time of the number of crossings of a diagram of L in
the following √ways. It can be shown by a skein relation that J2 (L; 1) = (−2)#L−1
and J2 (L; e2π −1/3 ) = 1, where #L denotes the number of components of L. It is
known that |J2 (L; −1)| is equal to the order of H1 (M2,L ) if its order is finite, and0 otherwise, where M2,L denotes
cover of S 3 branched along√ branched√ the double
L. It is known [49] that J2 (L; −1) = (− 2)#L−1 (−1)Arf(L) if Arf(L) exists, and√√ #L−1 √ dimH1 (M2,L ;Z/3Z)0 otherwise. It is known [43] that J2 (L; eπ −1/3 ) = ± −1
−3.
By using such topological interpretations, the Jones polynomial can be calculated
in polynomial time at 2nd, 3rd, 4th, 6th roots of unity.
When q is not such a root of unity, it is known [32, 61] that computing J2 (L; q)of an alternating link L is #P-hard.
The following remark is due to Tetsuya Ito.
Remark (T. Ito). It is known [30] that the invariants of Question 1.1 can be presented
by the colored Alexander invariants, which are defined by homological representations of the braid groups generalizing the Burau representation. By using theseinvariants, the invariants of Question 1.1 can be calculated in polynomial time, andQuestion 1.1 is solved aﬃrmatively.

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2

The minimal coloring number of Z-colorable links(Kazuhiro Ichihara2 and Eri Matsudo3 )The Fox n-coloring would be one of the most well-known invariants of knots andlinks (n ≥ 2 an integer). However, some of links are known to admit non-trivial Fox
n-colorings for every n ≥ 2, that is, the links with 0 determinants (see the remark
at the end of this section). For such a link, we can define a Z-coloring as follows,which is a natural generalization of the Fox n-coloring.
Let L be a link and D a regular diagram of L. A map γ from the set of the arcsof D to Z is called a Z-coloring on D if it satisfies the condition 2γ(a) = γ(b) + γ(c)at each crossing of D with the over arc a and the under arcs b and c. A Z-coloring
which assigns the same integer to all the arcs of the diagram is called the trivial
Z-coloring. A link is called Z-colorable if it has a diagram admitting a non-trivial
Z-coloring. (As usual, we call the integers appearing in the image of a Z-coloring
the colors.)
For a Z-colorable link L, the minimal coloring number of a diagram D of L isdefined as the minimal number of the colors for all non-trivial Z-colorings on D,
and the minimal coloring number of L is defined as the minimum of the minimalcoloring numbers of diagrams representing the link L.
The minimal numbers of colors for knots and links admitting Fox’s colorings
behave interestingly, and have been studied in detail recently. On the other hand,the following was shown by the second author in [47] and by Meiqiao Zhang, Xian’an
Jin and Qingying Deng in [62] independently, based on the result given in [27].
Theorem ([62], [47]) The minimal coloring number of any non-splittable Z-colorable
link is equal to 4.
We here remark that the proofs in both [47] and [62] are quite algorithmic, andso, the resultant diagrams in their proofs admitting a Z-coloring with four colors
are often very complicated.
In view of this, in [28], we have considered and studied the minimal coloring
numbers of minimal diagrams of Z-colorable links, that is, the diagrams representing
the link with least number of crossings.
Based on the results obtained in [28], the following problems can be considered.
Problem 2.1 (K. Ichihara, E. Matsudo). Determine the minimal coloring numberof minimal diagrams of Z-colorable torus links.
Remark. It is known that which torus links admit non-trivial Z-colorings. See [1] forexample to compute the determinants of torus links. In [28, Theorem 1.3], we showed
that, for even integer n > 2 and non-zero integer p, the torus link T (pn, n) has aminimal diagram admitting a Z-coloring with only four colors. We have studied
several other cases, but not obtained the complete classification yet.
2 Department of Mathematics, College of Humanities and Sciences, Nihon University, 3-25-40 Sakurajosui,
Setagaya-ku, Tokyo 156-8550, Japan
Email address: ichihara@math.chs.nihon-u.ac.jp
3 Graduate School of Integrated Basic Sciences, Nihon University, 3-25-40 Sakurajosui, Setagaya-ku, Tokyo 1568550, Japan
Email address: cher16001@g.nihon-u.ac.jp

3

Problem 2.2 (K. Ichihara, E. Matsudo). Determine the minimal coloring numberof minimal diagrams of Z-colorable pretzel links.
Remark. It is also known that which pretzel links admit non-trivial Z-colorings. See[14] to compute their determinants. In [28], we also obtained some results for such
links, but not obtained the complete classification yet.
Remark. A topological interpretation of a Z-coloring is a homomorphism π1 (S 3 −L) → Aut(Z), where we denote by Aut(Z) the subgroup of Map(Z, Z) generated
by fa (for a ∈ Z) given by fa (x) = 2a − x. This homomorphism naturally induces
a homomorphism H1 (M2,L ; Z) → 2Z, where M2,L denotes the double cover of S 3
branched along L. We note that Z naturally acts on the set of Z-colorings byadding a constant to all colors of a coloring. We have a natural bijection{}()
Z-colorings of a diagram of L /Z −→ Hom H1 (M2,L ; Z), 2Z .
The determinant of L is the determinant of a presentation matrix of H1 (M2,L ) (seee.g. [42]). Hence, when the determinant of L is 0, the rank of H1 (M2,L ) is positive,
and we have non-trivial Z-colorings of L.

3

Triangulations and the 3D index of cusped 3-manifolds(Neil Hoﬀman)
Let (M, T ) be an cusped, orientable 3-manifold and T be an ideal triangulationof M . We say T is 1-eﬃcient if the only embedded normal surfaces in T with
non-negative Euler characteristic are the boundary linking tori (made up solely of
triangular disks). In our context, we say a triangulation is 0-eﬃcient if there areno embedded normal S 2 or RP 2 . The concept of 0-eﬃcient and 1-eﬃcient triangulations was introduced by Jaco and Rubinstein [31] and they were able to showthat any triangulation of an irreducible manifold can be algorithmically simplified
to 0-eﬃcient one.
The 3D-index is an invariant introduced by Dimofte, Gaiotto and Gukov [15],which associated to a 1-eﬃcient ideal triangulation of a 3-manifold with torus boundary components. Furthermore, if two 1-eﬃcient triangulations are associated by a
2/3, 3/2, 0/2 or 2/0 Pachner move, then the 3D-indices of both triangulations are
identical by work of Garoufalidis, Hodgson, Rubinstein and Segerman [20]. This
motivates the following question which appears in [19] and in discussed in Section
12 of that paper:
Question 3.1 (S. Garoufalidis, C.D. Hodgson, N.R. Hoﬀman, J.H. Rubinstein [19]).
Given a cusped, atoroidal 3-manifold M . Are all 1-eﬃcient triangulations connectedby 2/3, 3/2, 0/2 or 2/0 Pachner moves?
It is worth mentioning that a more basic question is also appears to be open.
Question 3.2 (folklore). Given a cusped, irreducible 3-manifold M . Are all 1eﬃcient triangulations connected by 2/3, 3/2, 0/2 or 2/0 Pachner moves?
4Henry Segerman showed that triangulations without degree 1 edges are connectedin the Pachner graph [56]. The absence of degree 1 edges is a necessary condition
for a triangulation to be 0-eﬃcient.
There is also a version of the 3D index which associates to each peripheral curve
a formal Laurent series. Suggesting that closed manifolds may be assigned a 3D
index in this way.
Question 3.3 (N. Hoﬀman). Is this assignment a topological invariant? That isgiven two diﬀerent Dehn surgery presentations of the same manifold, will the 3D
index for each presentation be the same?
A lighter version of this question was asked by Dongmin Gang [18].
Conjecture 3.4 (D. Gang [18]). If the 3D index evaluated on a slope is an infinite
series starting with 1+· · · , if the Dehn filling corresponds to a hyperbolic 3-manifold
and the 3D index is 0, 1 or ∞ (does not converge), if M is non-hyperbolic. In
particular, if the Dehn filling is a lens space, then the 3D index is 0 .
We remark that Gang’s conjecture is supported by a number of a computations.
Finally, let infinite collection of Dehn fillings {Mγi } of a cusped manifold M suchthat the geometric limit of the {Mγi } is M .
Question 3.5 (N. Hoﬀman). Is there a normalization for the 3D index such thata sequence of 3D indices computed on the {Mγi } converges (term-wise) to the 3D
index of M ?

4

Destabilized Heegaard surfaces of 3-manifolds(Yeonhee Jang, Tsuyoshi Kobayashi, Makoto Ozawa and Kazuto Takao)
We briefly recall some of the standard terminology and facts. A Heegaard surface
of a 3-manifold is an embedded closed surface which divides the manifold into two
handlebodies. It is an important fact that every closed orientable 3-manifold has a
Heegaard surface. For a given Heegaard surface, we can construct a new Heegaard
surface of the same 3-manifold by adding a canceling pair of handles. A Heegaard
surface is said to be destabilized if it cannot be produced by this construction.
The following is a general problem.
Problem 4.1. Classify the Heegaard surfaces of each closed orientable 3-manifold.
In particular, classifying the destabilized ones is an essential part.
We briefly review some of the abundant known results for the above problem.
Waldhausen [60] showed the uniqueness of destabilized Heegaard surfaces of the 3sphere, and Bonahon and Otal [10, 11] showed that of lens spaces. Hence, if there isa Heegaard surface of genus 0 or 1, then the destabilized ones of the 3-manifold are
unique. On the other hand, Casson–Gordon (see [48, 54]) and Moriah–Schleimer–
Sedgwick [48] gave families of 3-manifolds each of which has infinitely many destabilized Heegaard surfaces of pairwise distinct genera. The minimal genus of Casson–
Gordon’s Heegaard surfaces is 4, and that of Moriah–Schleimer–Sedgwick’s is 3,
5though we do not know whether they attain the minimums of the manifolds. We
also refer the reader to [53, 58] for a theorem on the stable equivalence, to [33, 40]
for theorems on the finiteness at any given genus, to [13, 36, 41] for classification
algorithms, and to [25, 35, 38, 45, 55] for relevant examples.
The following is one of various open problems suggested by the above results.
Problem 4.2. Prove or disprove the existence of a 3-manifold which has a Heegaard
surface of genus 2, and infinitely many destabilized ones of pairwise distinct genera.

5

Simplified trisections of 4-manifolds(Osamu Saeki4 )A trisection of a 4-manifold was introduced by Gay–Kirby [21], and it is expected
to be a generalization of a Heegaard splitting of a 3-manifold. Roughly speaking,a trisection of a 4-manifold M is a decomposition M = M1 ∪ M2 ∪ M3 such thatfor a fixed non-negative integer ℓ, Mi is diﬀeomorphic to ♮ℓ (S 1 × B 3 ) (4-dimensional
handlebody) for i = 1, 2, 3, and M1 ∩ M2 ∩ M3 is a closed orientable surface of genus
g, and Xk = (Mk ∩ Mi ) ∪ (Mk ∩ Mj ) is a 3-manifold having a Heegaard splitting ofgenus g as the union of Mk ∩ Mi and Mk ∩ Mj for {k, i, j} = {1, 2, 3}. As a Heegaard
splitting of a 3-manifold X can be studied by using a Morse function X → R, a
trisection of a 4-manifold M can be studied by using a Morse 2-function M → R2 .
Let M be a smooth closed connected oriented 4-manifold. A smooth map f :M → R2 is a trisected Morse 2-function (or a trisection map) if it has only fold andcusp singularities and it satisfies the following conditions (see [21, 8, 9] and Fig. 1):(1) its image is diﬀeomorphic to a 2-disk, denoted by D2 ,(2) it has a single definite fold circle mapped diﬀeomorphically onto the boundary∂D2 of D2 ,(3) D2 \ {p} can be nonsingularly foliated by rays from a regular value p to ∂D2 ,each intersecting the indefinite fold image always in the direction of index-2
handle attachments,(4) three of these rays split D2 into three sectors, where there is at most one cusp
on each singular arc image in a sector,(5) the total number of cusps in the sectors are equal,(6) the singular arcs with cusps are situated inside.
The number g of indefinite fold arcs (possibly with a cusp) in each sector is called
the genus of the trisection. Note that f −1 (p) is a closed orientable surface of genus
g for the regular value p ∈ D2 .
It is known that for every trisection decomposition of M , there is a trisected
Morse 2-function f : M → R2 which gives the given trisection decomposition.
4 Institute of Mathematics for Industry, Kyushu University, Motooka 744, Nishi-ku, Fukuoka 819-0395, Japan
Email address: saeki@imi.kyushu-u.ac.jp

6pFigure 1: The figure on the left depicts the image of a trisected Morse 2-function. In each box
there is an arbitrary Cerf graphic as in the right [21, §3]. The three half lines divide the image into
three parts and their inverse images correspond to 4-dimensional handlebodies giving a trisection
decomposition.

As far as the author knows, the following problem is still open.
Problem 5.1 (R.I. Baykur, O. Saeki [8]). Non-isotopic trisected Morse 2-functions
may yield equivalent trisection decompositions. Describe the necessary and suﬃcientconditions for two trisected Morse 2-functions to give equivalent trisection decompositions.
A trisection is said to be simplified if there exists an associated trisection map
such that the restriction to its singular locus is an embedding [8, 9] (see Fig. 2).
This is in great contrast with a general trisection map (see Fig. 1), which has Cerfboxes in between the three sectors of the image 2-disk, where folds can cross eachother arbitrarily (and therefore, the images of some indefinite fold circles might windaround p multiple times).
Note that Hayano [26] has studied the condition for a given trisection decomposition to be equivalent to a simple one. He has also classified genus-2 simplified
trisections.
The following manifolds are known to admit genus-3 simplified trisections [9]:S 4 , connected sums of CP2 , CP2 and S 1 × S 3 with three summands, connected sumof either one of these manifolds with S 2 × S 2 , Pao’s manifolds Ln , L′n [51]. One
can similarly get genus-4 examples on connected sums of lower genera trisections
on these standard 4-manifolds. In addition, we have the irreducible examples on
L(p, q)-bundles and (S 1 × S 2 )-bundles over S 1 , which include S 2 -bundles over the
2-torus T 2 and the Klein bottle Kb.
The following two problems have been posed in [9].
Problem 5.2 (R.I. Baykur, O. Saeki [9]). Classify 4-manifolds that admit simplified
genus-3 trisections. Is there any 4-manifold, other than the ones mentioned above,
which admits genus-3 simplified trisections? How about genus-4?
7

Figure 2: Singular value of a simplified trisected Morse 2-function

Question 5.3 (R.I. Baykur, O. Saeki [9]). Is there any 4-manifold which admits atrisection, but not a simplified one of the same genus? Defining the minimal trisection genus (resp. minimal simplified trisection genus) of a 4-manifold M as thesmallest genus of a trisection (resp. simplified trisection) on M , one can equivalently ask if there is a 4-manifold whose trisection genus is strictly smaller than its
simplified trisection genus.
The two genera are equal for all of the 4-manifolds with (simplified) trisectionsof genus g ≤ 4 mentioned above. The answer to the analogous question for broken
Lefschetz fibrations versus simplified broken Lefschetz fibrations is positive [7].
Heegaard splittings of a closed oriented 3-manifold X are closely related with a
Morse function g : X → R. Based on this, Johnson [34] and Takao [59] used genericmaps into R × R = R2 for comparing two Heegaard splittings of a given 3-manifold.
Question 5.4 (O. Saeki). Can we use a similar idea for comparing two trisectionsof a given 4-manifold to get some information on the relationship between the twotrisections?
It is well known that any two Heegaard splittings of a given 3-manifold are
related by a sequence of stabilizations. As an analog, it is shown [21] that any twotrisections of a given 4-manifold are related by a sequence of “stabilizations”. In[21], this is proved by using handle decompositions of a 4-manifold and stabilizationsof Heegaard splittings of 3-manifolds.
Problem 5.5 (D. Gay, R. Kirby [21]). Find a singularity theoretical proof of theuniqueness of trisections (up to stabilization) on a given 4-manifold.
See [21, Remark 15] for an approach to this problem.

8

6

Virtual embeddings between mapping class groups of surfaces(Takuya Katayama)
Let Σbg,p be a compact orientable surface of genus g with p marked points andb boundary components. The mapping class group (or Teichmüller modular group)Mod(Σbg,p ) of Σbg,p is the group of orientation-preserving homeomorphisms of Σbg,pfixing the set of marked points setwise and fixing the boundary pointwise, up to
isotopy relative to the marked points and the boundary. In particular, Mod(Σ10,n ) isnaturally identified with the nth braid group Bn .
Homomorphisms between mapping class groups have been studied in many researches. A typical construction of such a homomorphism is induced by an embedding of a surface, i.e., a combination of forgetting marked points, deleting boundary components, and subsurface embeddings; see [4]. Another typical constructionof such a homomorphism is induced by a covering of a surface. For example, anatural double branched covering Σ02,0 → Σ00,6 induces a surjective homomorphism
Mod(Σ02,0 ) → Mod(Σ00,6 ), whose kernel is a cyclic group of order 2; see [46]. Further,natural double coverings Σ1g,0 → Σ10,2g+1 and Σ2g,0 → Σ10,2g+2 induce injective homomorphisms B2g+1 → Mod(Σ1g,0 ) and B2g+2 → Mod(Σ2g,0 ); see [16, Section 9.4]. The
topic of homomorphisms between mapping class groups is related to the rigidity ofstructures on surfaces; see [5] for a survey on this topic.
Moreover, injective homomorphisms between mapping class groups have also beenstudied in some researches. By modifying the above mentioned homomorphisms of
B2g+1 and B2g+2 , we can obtain injective homomorphisms B2g+1 → Mod(Σ0g+1,0 )and B2g+2 → Mod(Σ0g+1,0 ); see [24, 37]. Further, it is known [3] that, for any g ≥ 2,
there exist g ′ > g and an injective homomorphism Mod(Σ0g,0 ) → Mod(Σ0g′ ,0 ). On
the other hand, it is known [24] that, when g ≥ 3 and g > g ′ , there are no nontrivial
homomorphisms Mod(Σ0g,0 ) → Mod(Σ0g′ ,0 ).
We say that a group H is embedded in another group G if there exists an injective
homomorphism from H to G. Further, as in [6], we say that H is virtually embedded
in G if a finite index subgroup of H is embedded in G. We note that the compositionof virtual embeddings is a virtual embedding.
As an example of a virtual embedding, we can show that Mod(Σ02,0 ) is virtuallyembedded in Mod(Σ00,6 ), as follows. As mentioned above, a double covering Σ02,0 →Σ00,6 induces a homomorphism Mod(Σ02,0 ) → Mod(Σ00,6 ) whose kernel is a cyclic
group of order 2 (generated by the hyperelliptic involution ι). Since Mod(Σ02,0 ) is
residually finite (see [16, Section 6.4]), we can find a finite index subgroup H of
Mod(Σ02,0 ) such that the restriction map H → Mod(Σ00,6 ) is injective, by showingthat its kernel {1, ι} ∩ H is trivial. Hence, Mod(Σ02,0 ) is virtually embedded inMod(Σ00,6 ). This construction is a topological interpretation of a virtual embedding.
See also [57, Theorem 2] for a preceding result on virtual embeddings.
Problem 6.1 (T. Katayama). Given g ≥ 2, determine whether Mod(Σ0g,0 ) is virtu′
ally embedded in Mod(Σbg′ ,p′ ).
9Remark 1. If g was 1, we can show that it is relatively easy to make a virtual
embedding of Mod(Σ01,0 ) to a mapping class group, as follows. It is known, see e.g.
[44, Proposition 4.4.2], that Mod(Σ01,0 ) is SL(2; Z) and it has the rank 2 free group asa finite index subgroup. Further, we can embed the rank 2 free group into almost all
mapping class groups, by considering Dehn twists along two essential simple closed
curves with at least two crossings; see [29]. Hence, if g was 1, Problem 6.1 would be
relatively easy.
Remark 2. When g = 2, we can show that Mod(Σ02,0 ) is virtually embedded inMod(Σ0g′ ,0 ) for any g ′ ≥ 3, as follows. As mentioned above, Mod(Σ02,0 ) is virtuallyembedded in Mod(Σ00,6 ). Further, by Remarks 2A and 2B below, Mod(Σ00,6 ) isvirtually embedded in Mod(Σ0g′ ,0 ) for any g ′ ≥ 3. Hence, Mod(Σ02,0 ) is virtuallyembedded in Mod(Σ0g′ ,0 ) for any g ′ ≥ 3.
Therefore, when g = 2, g ′ ≥ 3 and p′ + b′ = 0, Problem 6.1 is solved aﬃrmatively.
′
It is not known whether Mod(Σ0g,0 ) is virtually embedded in Mod(Σbg′ ,p′ ), when g ≥ 3
or p′ + b′ ≥ 1.
Remark 2A. We can show that Mod(Σ00,6 ) is virtually embedded in B5 , as follows. It is known [12, Theorem 10] that the pure braid group P5 is isomorphic
to PMod(Σ00,6 ) × Z, where PMod(Σ00,6 ) is the finite index subgroup of Mod(Σ00,6 )fixing the marked points. Since there is an embedding PMod(Σ00,6 ) → P5 → B5 ,Mod(Σ00,6 ) is virtually embedded in B5 .
Remark 2B. We can show that B5 is embedded in Mod(Σ0g,0 ) for any g ≥ 3, as
follows. As mentioned before, there is an embedding B2g+1 → Mod(Σ0g+1,0 ). Hence,
when g ≥ 3, there is an embedding B5 → B2g−1 → Mod(Σ0g,0 ). Therefore, B5 isembedded in Mod(Σ0g,0 ) for any g ≥ 3.
Remark 3. It is known [37, Corollary 1.7] that, when g, g ′ ≥ 2, Mod(Σ0g,p ) is virtuallyembedded in Mod(Σ0g′ ,p′ ), only if 3g + p ≤ 3g ′ + p′ and 2g + p ≤ 2g ′ + p′ .
It follows from [4, Corollary 1.2] that, if 6 ≤ g < g ′ ≤ 2g − 2, Mod(Σ0g,0 ) is notembedded in Mod(Σ0g′ ,0 ).
It is known [4, Proposition 7.1] that, if g ≥ 3 and g > g ′ , any homomorphism′
Mod(Σbg,p ) → Mod(Σbg′ ,p′ ) is trivial.
Conjecture 6.2 (T. Katayama). Suppose that g ≥ 2. If Mod(Σbg,p ) is virtuallyembedded in Bn for some n, then (g, p, b) = (2, 0, 0).
When (g, p, b) = (2, 0, 0), we note that Mod(Σ02,0 ) is virtually embedded in B5 by
Remarks 2 and 2A of Problem 6.1.
In case there is a counter-example to Conjecture 6.2, the target braid group does
not admit faithful C ∞ action on R even virtually, and is not “virtually special” (i.e.,
no finite index subgroup is embedded in a right-angled Artin group, see [6, Question2]).
Problem 6.3 (T. Katayama). Find a pair (H, ϕ) of a finite index subgroup H of′
Mod(Σbg,p ) and a homomorphism ϕ : H → Mod(Σbg′ ,p′ ) with “small kernel”.
10Remark. Some “surface manipulations” induce homomorphisms with “small kernels”(see [3], [4], [16, Chapter 3], [46] and [52]). Are there any other homomorphisms
with “small kernels”?

7

Totally real immersions and embeddings of n-manifolds
into Cn(Naohiko Kasuya5 )Let M n be a closed orientable n-manifold and f : M n → Cn be an immersion.
A point p ∈ M n is said to be a complex tangent if dfp (Tp M n ) contains a complex
line. An immersion is said to be totally real if it has no complex tangent. It follows
from Thom’s transversality theorem that the set of complex tangents of a genericimmersion f : M n → Cn is empty or forms a closed (n−2)-dimensional submanifold.
For totally real immersions and embeddings, the following theorems are known.
These are called the h-principle for totally real immersions and embeddings.
Theorem (Gromov [22], Lees [39]). An n-manifold M n admits a totally real immersion into Cn if and only if the complexified tangent bundle CT M n is trivial.
Theorem (Gromov [23], Forstnerič [17]). Let M n be a closed orientable n-manifoldwith n ≥ 3. Then, M n admits a totally real embedding into Cn if and only if it
admits a totally real immersion into Cn which is regularly homotopic to an embedding.
In the following, we consider the case where n = 3. It follows from the hprinciple that any closed orientable 3-manifold admits a totally real embedding into
C3 . However, there are few explicit examples of totally real embeddings. The
following example is due to Ahern and Rudin.
Example (Ahern-Rudin [2]). {Let P (z, w) = z̄ w̄(|w|2}+ i|z|2 ). We consider the 3sphere as the unit sphere S 3 = (z, w) |z|2 +|w|2 = 1 ⊂ C2 . Then, the embeddingf : S 3 → C3 defined by()
f (z, w) = z, w, P (z, w)
is a totally real embedding.
Inspired by their example, Forstnerič constructed some other examples of totallyreal embeddings of quotients of the 3-sphere, including RP 3 . But we want more
examples.
Problem 7.1 (N. Kasuya). Construct interesting examples of totally real embeddings of 3-manifolds into C3 (without using the h-principle). For example, can a
Haefliger knot be explicitly realized as a totally real submanifold of C3 ?

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