On a construction of stable maps from 3-manifolds into surfaces

TL;DR

This paper presents a visual method to construct stable maps from 3-manifolds, specifically from the 3-sphere and other closed orientable 3-manifolds, into surfaces, with controlled singularities and fiber types.

Contribution

It introduces a new visual construction technique for stable maps from 3-manifolds into surfaces, linking links in the 3-sphere to specific stable map properties.

Findings

Constructed stable maps with no cusp points

Set of definite fold points coincides with the given link

Obtained stable maps into the 2-sphere for all closed orientable 3-manifolds

Abstract

For any link in the $3$-sphere, we give a visual construction of a stable map $f$ from the $3$-sphere into the real plane enjoying the following properties; $f$ has no cusp point, the set of definite fold points of $f$ coincides with the given link and $f$ only has certain type of fibers containing two indefinite fold points. As a corollary, we obtain a similar stable map from every closed orientable $3$-manifold into the $2$-sphere.

Full text

1. Introduction

There have been many studies focusing on stable maps from a closed orientable smooth $3$-manifold to the Euclidean plane $\mathbb{R}^{2}$. See [1, 3, 4, 5, 8, 12, 13, 14, 15, 16, 17, 18, 19], for example. The definition of stable maps will be introduced in Section 2. In this paper, all manifolds and maps are assumed to be differentiable of class $C^{\infty}$ unless otherwise indicated.

Let $M$ be a closed orientable $3$-manifold and let $L$ be any given link in $M$. It is shown by Saeki [14, 15, Corollary 6.3] that there exists a stable map $f:M\to\mathbb{R}^{2}$ such that the set of singular points $S(f)$ is equal to $L$ if and only if $[L]_{2}=0$ in $H_{1}(M;\mathbb{Z}_{2})$. For such a stable map $f$, it is known that the points in $S(f)$ are classified into definite fold points, indefinite fold points, and cusp points. Further, it is known that singular fibers of $f$ containing two indefinite fold points are classified into type $\mathrm{I\hskip-1.2ptI^{2}}$ and type $\mathrm{I\hskip-1.2ptI^{3}}$ (Figure 1). See [7, 8], for example. Then, in [3, Corollary 3.7], Ishikawa and Koda showed that there exists a stable map $f\colon M\to\mathbb{R}^{2}$ without cusp points such that $f$ has no singular fibers of type $\mathrm{I\hskip-1.2ptI^{3}}$ and the set of definite fold points $S_{0}(f)$ contains the given link $L$.

In this paper, we give a partial refinement of these results. It is well known that any oriented link in the $3$-sphere $S^{3}$ is represented as the closure of a braid, and any braid is presented by a braid word with the generators $\sigma_{i}^{\pm 1}$, which are shown in Figure 2. See [9, Chapter 1], for example. We denote the set of type $\mathrm{I\hskip-1.2ptI^{2}}$ singular fibers of $f$ by $\mathrm{I\hskip-1.2ptI^{2}}(f)$, and we denote the cardinality of a finite set $S$ by $|S|$.

Theorem 1.1**.**

For any link $L$ in $S^{3}$, there exists a stable map $f:S^{3}\to\mathbb{R}^{2}$ without cusp points such that $f$ has no singular fibers of type $\mathrm{I\hskip-1.2ptI^{3}}$ and $S_{0}(f)$ is isotopic to $L$. Moreover, if $L$ is represented as the closure of an $n$-braid $\sigma_{z_{1}}^{m_{1}}\sigma_{z_{2}}^{m_{2}}\cdots\sigma_{z_{l}}^{m_{l}}$ with $n\geq 2$, $1\leq z_{i}\leq n-1$, $z_{i-1}\neq z_{i}$ and $m_{i}$ non-zero integers, then $|\mathrm{I\hskip-1.2ptI^{2}}(f)|=2(l-|X|)$ holds with $X=\{i\mid z_{i}=1,~1\leq i\leq l\}$.

Remark 1.2*.*

While Ishikawa and Koda [3] proved their theorems using $4$-dimensional topology, we establish the theorem above using only $3$-dimensional techniques. Our construction is similar to the method developed in [2].

It is well known that every closed orientable $3$-manifold can be obtained from $S^{3}$ by integral Dehn surgery on a link in $S^{3}$. See Section 3 for further details. Together with this, we have the following as a corollary of Theorem 1.1.

Theorem 1.3**.**

Let $M$ be a closed orientable $3$-manifold. Then, there exists a stable map $f_{0}$ from $M$ into $S^{2}$ without cusp points, definite fold points and singular fibers of type $\mathrm{I\hskip-1.2ptI^{3}}$. In particular, if $M$ is obtained by an integral Dehn surgery on a link $L$ in $S^{3}$ which is represented as the closure of a pure braid $b=\sigma_{z_{1}}^{m_{1}}\sigma_{z_{2}}^{m_{2}}\cdots\sigma_{z_{i}}^{m_{i}}\cdots\sigma_{z_{l}}^{m_{l}}$, then $|\mathrm{I\hskip-1.2ptI^{2}}(f)|=2(l-|X|)$ holds with $X=\{i\mid z_{i}=1,~1\leq i\leq l\}$.

We remark that, under different assumptions and conditions, there are other studies concerning the construction of stable maps on $3$-manifolds obtained by Dehn surgery. See [4, Theorem 1.2], for example.

2. A construction of stable maps from $S^{3}$ into the plane

In this section, we give a proof of Theorem 1.1. Before proceeding to the proof, we introduce several definitions and known facts.

For smooth manifolds $M$ and $N$, let $C^{\infty}(M,N)$ be the set of smooth maps from $M$ to $N$ with the Whitney topology. A smooth map $f:M\to N$ is called a stable map if there exists a neighborhood $U_{f}$ of $f$ in $C^{\infty}(M,N)$ such that, for any map $g$ in $U_{f}$, there are diffeomorphisms $\Phi:M\to M$ and $\phi:N\to N$ satisfying $g=\phi\circ f\circ\Phi^{-1}$.

In the case where the source and target dimensions are $3$ and $2$, a characterization of stable maps is given as follows ([8]). For a $3$-manifold $M$, a smooth map $f:M\to\mathbb{R}^{2}$ is a stable map if and only if $f$ is locally described as in one of the following forms:

1. (1)

$(u,x,y)\mapsto(u,x)$, 2. (2)

$(u,x,y)\mapsto(u,x^{2}+y^{2})$, 3. (3)

$(u,x,y)\mapsto(u,x^{2}-y^{2})$, 4. (4)

$(u,x,y)\mapsto(u,y^{2}+ux-x^{3})$,

and $f$ globally satisfies

1. (5)

$f^{-1}(f(p))\cap S(f)=\{p\}$ for a cusp point $p$, 2. (6)

except for cusp points, the restriction of $f$ to $S(f)$ is an immersion with only normal crossings.

The points around which $f$ is described as (2), (3), and (4) are the singular points of $f$, and they are called definite fold points, indefinite fold points, and cusp points, respectively. The set of definite fold points and indefinite fold points is denoted by $S_{0}(f)$ and $S_{1}(f)$, respectively. Consequently, to prove Theorem 1.1, it suffices to construct a smooth map satisfying these conditions with the described properties. Throughout the following, we denote the boundary of a manifold $N$ by $\partial N$.

Proof of Theorem 1.1..

Let $L$ be a link in $S^{3}$. Suppose that $L$ is represented by the closure of a braid $b$ of $n$ strings $(n\geq 2)$ given as a braid word $\sigma_{z_{1}}^{m_{1}}\sigma_{z_{2}}^{m_{2}}\cdots\sigma_{z_{l}}^{m_{l}}$, where $1\leq z_{i}\leq n-1$, $z_{i-1}\neq z_{i}$, $m_{i}$ is non-zero integers for $i\in\{1,2,\ldots,l\}$, and $l\geq 1$.

We will obtain a stable map $S^{3}\to\mathbb{R}^{2}$ by gluing several smooth maps on submanifolds of $S^{3}$. To construct these smooth maps, we make the following preparations. We decompose $S^{3}$ into two solid tori $V_{1}$ and $V_{2}$ as $S^{3}=V_{1}\cup_{\phi}V_{2}$. Here, $\phi$ denotes a diffeomorphism from $\partial V_{1}$ onto $\partial V_{2}$ such that a meridian of $V_{1}$ corresponds to a longitude of $V_{2}$ by $\phi$. (A meridian of $V_{1}$ is $\partial\mathbb{D}^{2}\times\{*\}\subset\mathbb{D}^{2}\times S^{1}=V_{1}$ and a longitude of $V_{2}$ is $\{*\}\times S^{1}\subset\mathbb{D}^{2}\times S^{1}=V_{2}$.) We further decompose $V_{1}$ into $N_{1}$ and $N_{2}$, each of which is diffeomorphic to $\mathbb{D}^{2}\times[0,l]$. We isotope $L$ so that $L$ is contained in $V_{1}$ and $L\cap N_{1}$ corresponds to the braid $b$. Next, we decompose $N_{1}$ into $l$ solid cylinders $W_{1},\ldots,W_{l}$, where $W_{i}$ is regarded as $\mathbb{D}^{2}\times[i-1,i]$ $(1\leq i\leq l)$ such that $L\cap W_{i}$ corresponds to $\sigma_{z_{i}}^{m_{i}}$ as shown in Figure 3. Let $A$ be an annular region in $\mathbb{R}^{2}$ and $E$ a disk in $\mathbb{R}^{2}$ such that the intersection $A\cap E=\partial E$. Also, we decompose $A$ into two rectangular regions $R_{1}$ and $R_{2}$ by $\gamma_{1},\gamma_{l+1}$. Moreover, we decompose $R_{1}$ into $l$ rectangular regions by arcs $\gamma_{2},\ldots,\gamma_{l}$ as shown in Figure 4.

Let $F_{i}$ be the disk $\mathbb{D}^{2}\times\{i-1\}$ in $N_{1}=W_{1}\cup\cdots\cup W_{l}$ $(1\leq i\leq l+1)$. Note that $F_{i}=W_{i-1}\cap W_{i}$ $(2\leq i\leq l)$. For each $i$, let $\psi_{i}:F_{i}\to\gamma_{i}$ be the map which is obtained by regarding $F_{i}$ as $\mathbb{D}^{2}$ and considering a smooth height function on $\mathbb{D}^{2}$ shown in Figure 5. Precisely, $\psi_{i}$ is the composite map of a diffeomorphism $F_{i}\to\mathbb{D}^{2}$ and a smooth height function on $\mathbb{D}^{2}$ shown in Figure 5. In Figure 5, the fat points on $F_{i}$ depict the local maxima of $\psi_{i}$. For the images of saddle points of $\psi_{i}$, their preimages on $F_{i}$ are figure-eight fibers under $\psi_{i}$. The Reeb graph $\tau_{i}$ is shown in the center of Figure 5. The boundary $\partial F_{i}$ is mapped to an endpoint of $\tau_{i}$.

For $i$ with $\sigma_{z_{i}}=\sigma_{1}$, we construct a smooth map $\Psi_{i}:W_{i}\to T_{i}$ obtained from an isotopy between $\psi_{i}$ and $\psi_{i+1}$ as follows. Let $P$ be the highest saddle point on $F_{i}$ and $\alpha$ a left-open interval in $\gamma_{i}$ containing $\psi_{i}(P)$ and $\max\mathrm{Im}\psi_{i}$. Then, the isotopy $F_{i}\times[0,1]\to F_{i}$ is given by $m_{i}$ half-twists on the connected component of the preimage of $\alpha$ containing $P$. See Figure 6 for an illustration of $\Psi_{i}$ when $m_{i}$ is even. Note that the image of definite fold points, i.e., $L\cap W_{i}$, has no normal crossings in this case. When $m_{i}$ is odd, the image of a surface obtained from $F_{i}$ via the isotopy is shown in Figure 7 (left). Further, we deform the height function by another isotopy which is defined by switching the heights of the two leftmost local maxima, as shown in Figure 7. This induces the switching of the images of the two uppermost maxima of $\mathbb{D}^{2}$ in the target. Thus, when $m_{i}$ is odd, a smooth map $\Psi_{i}$ constructed by connecting the two isotopies is illustrated in Figure 8. Note that the image of definite fold points, i.e., $L\cap W_{i}$, has a normal crossing when $m_{i}$ is odd. Except for this switching, the image of definite fold points has no normal crossings in this case. Furthermore, regardless of the parity of $m_{i}$, the constructed smooth map $\Psi_{i}$ has only definite and indefinite fold points, and has no cusp points.

We make further remarks on Figures 6 and 8. The arcs of $L\cap W_{i}$ correspond to the segments parallel to the upper edge of $T_{i}$ by $\Psi_{i}$. As noted above, when $m_{i}$ is odd, the segments contain a normal crossing. On $T_{i}$, thick arcs represent parts of the image of indefinite fold points. The boundaries $\partial F_{i}$ and $\partial F_{i+1}$ correspond to the endpoints of the lower edge of $T_{i}$.

For $i$ with $\sigma_{z_{i}}\neq\sigma_{1}$, to construct a smooth map $\Psi_{i}:W_{i}\to T_{i}$, we first consider the deformation between $\psi_{i}$ and $\psi_{i+1}$ illustrated in Figure 9. In Figure 9, the deformation starts at the top corresponding to $\psi_{i}$, goes down to obtain another smooth map $\psi_{i}^{\prime}:F_{i}^{\prime}\to\gamma_{i}^{\prime}$, goes down to obtain another smooth map $\psi_{i}^{\prime\prime}:F_{i}^{\prime\prime}\to\gamma_{i}^{\prime\prime}$, and gets back conversely to the top, corresponding to $\psi_{i+1}$. We denote the intermediate disk (middle) by $F_{i}^{\prime}$ and the intermediate disk (bottom) by $F_{i}^{\prime\prime}$. Note that during the deformation in Figure 9, the singular points and singular fibers move on the disks. The movement can be understood by observing the heights of two adjacent saddle points. At $F_{i}^{\prime}$, the heights of these two adjacent saddle points lie at the same level. At $F_{i}^{\prime\prime}$, the relative positions of these two adjacent saddle points are switched.

We secondly consider an isotopy from $\psi_{i}^{\prime}$ to $\psi_{i}^{\prime}$ through $\psi_{i}^{\prime\prime}$. The isotopy is determined by $m_{i}$ half-twists in the same way as in the case $z_{i}=1$. When $m_{i}$ is even, we plug in the isotopy during the deformation from $\psi_{i}^{\prime}$ to $\psi_{i}^{\prime}$. Then, a smooth map $\Psi_{i}:W_{i}\to T_{i}$ is obtained by the deformation with isotopy, which is illustrated in Figure 12. In Figure 12, $F_{i}^{\prime}$ appears twice in $W_{i}$ on both sides of the crossings of $L$ corresponding to $\sigma_{z_{i}}^{m_{i}}$. On each $F_{i}^{\prime}$, there is a singular fiber of type $\mathrm{I\hskip-1.2ptI^{2}}$. When $m_{i}$ is odd, we construct the smooth map $\Psi_{i}$ by combining two isotopies. The first is the isotopy during the deformation from $\psi_{i}^{\prime}$ to $\psi_{i}^{\prime}$, and the second is defined by switching the heights of the $z_{i}$-th and $(z_{i}+1)$-th local maxima from the left in Figure 11. Similarly to the case where $z_{i}=1$, this switching creates a normal crossing in the image of $S_{0}(f)$. Consequently, as illustrated in Figure 11, the map $\Psi_{i}$ is almost identical to the case where $m_{i}$ is even, except for the presence of this normal crossing. Regardless of the parity of $m_{i}$, the constructed smooth map $\Psi_{i}$ has definite fold points, indefinite fold points, and only two singular fibers of type $\mathrm{I\hskip-1.2ptI^{2}}$, and has no cusp points.

By connecting $\Psi_{1}:W_{1}\to T_{1},\ldots,\Psi_{l}:W_{l}\to T_{l}$ along $\psi_{2}:F_{2}\to\gamma_{2},\ldots,\psi_{l}:F_{l}\to\gamma_{l}$, we obtain a smooth map $N_{1}\to R_{1}$. We connect this map with a smooth map $N_{2}\to R_{2}$ naturally induced from $\psi_{1}:F_{1}\to\gamma_{1}$ and $\psi_{l+1}:F_{l+1}\to\gamma_{l+1}$ by the product structure of $N_{2}$. As a result, a smooth map $g_{1}:V_{1}\to A$ is obtained. Further, connecting $g_{1}$ and the natural projection $g_{2}:V_{2}\to E$, we finally obtain a smooth map $f:S^{3}\to A\cup E\subset\mathbb{R}^{2}$. By construction, this map $f$ has no cusp points, and its singular set consists of definite and indefinite fold points. Also, $f$ satisfies the global conditions (5) and (6). Moreover, the set of definite fold points forms a link isotopic to $L$ and $f$ has only singular fibers of type $\mathrm{I\hskip-1.2ptI^{2}}$. Consequently, $f:S^{3}\to\mathbb{R}^{2}$ is a stable map without cusp points such that $f$ has no singular fibers of type $\mathrm{I\hskip-1.2ptI^{3}}$ and $S_{0}(f)$ is isotopic to $L$.

Furthermore, $|\mathrm{I\hskip-1.2ptI^{2}}(f)|=2(l-X)$ holds by the construction of $f$, since $L$ is represented as the closure of an $n$-braid $\sigma_{z_{1}}^{m_{1}}\sigma_{z_{2}}^{m_{2}}\cdots\sigma_{z_{l}}^{m_{l}}$ with $n\geq 2$, $1\leq z_{i}\leq n-1$, $z_{i-1}\neq z_{i}$, $m_{i}\neq 0$.

∎

Example 2.1*.*

Following the construction in the proof above, we provide a stable map $f:S^{3}\to\mathbb{R}^{2}$ shown in Figure 13 such that $S_{0}(f)$ is isotopic to $K=10_{128}$ in the standard knot table. It is known that $K$ is represented by a $4$-string braid with the braid word $\sigma_{1}^{-3}\sigma_{2}^{-1}\sigma_{1}^{-2}\sigma_{2}^{-2}\sigma_{3}^{-1}\sigma_{2}\sigma_{3}^{-1}$. For details, see [10]. In Figure 13 (right), the image of $S^{3}$ by $f$ is a disk in $\mathbb{R}^{2}$ such that $f(S_{1}(f))$ is the thick immersed curve in the disk. Also, $S_{0}(f)$, which is isotopic to $K$, is the immersed curve depicted in Figure 13 (left). In Figure 13 (right), the box $(\text{B})$ represents a specific crossing pattern as illustrated in Figure 14. This $f$ has no singular fibers of type $\mathrm{I\hskip-1.2ptI^{3}}$. Only some of the singular fibers are shown in the figure.

3. a stable map on a $3$-manifold

In this section, we give a proof of Theorem 1.3. Before proceeding to the proof of Theorem 1.3, we recall the definition of Dehn surgery. Let $K$ be a knot in $S^{3}$, and let $E(K)$ denote the exterior of $K$, defined as $S^{3}\setminus\text{int}\,N(K)$. The isotopy class of an unoriented simple closed curve on $\partial E(K)$ is called a slope. By using the standard meridian-longitude system, the set of slopes on $\partial E(K)$ is identified with $\mathbb{Q}\cup\{1/0\}$. A $3$-manifold is obtained by gluing a solid torus $V$ to $E(K)$ so that the meridian of $V$ is identified with a curve of slope $p/q$ on $\partial E(K)$. This operation is called $p/q$-Dehn surgery. Similarly, Dehn surgery on a link is defined for any link. In the following, we consider the case of $p$-Dehn surgery (i.e., $q=1$), which we call integral Dehn surgery. See [9, Chapter 12] for details.

Proof of Theorem 1.3..

It is known that every closed orientable $3$-manifold is obtained from $S^{3}$ by integral Dehn surgery on the closure of a pure braid. See [6, p.115], for example. Suppose that a link $L=K_{1}\cup\cdots\cup K_{l}$ in $S^{3}$ is represented as the closure of a pure braid $b=\sigma_{z_{1}}^{m_{1}}\sigma_{z_{2}}^{m_{2}}\cdots\sigma_{z_{l}}^{m_{l}}$. Note that the number of components of $L$ is equal to the number of strands of $b$. Then, by Theorem 1.1, we have a stable map $f:S^{3}\to\mathbb{R}^{2}$ such that $f$ has no cusp points and no singular fibers of type $\mathrm{I\hskip-1.2ptI^{3}}$, $S_{0}(f)$ is isotopic to $L$, and $|\mathrm{I\hskip-1.2ptI^{2}}(f)|=2(l-|X|)$ holds with $X=\{i\mid z_{i}=1,~1\leq i\leq l\}$. Note that for each $i$, the image $f(K_{i})$ of $K_{i}\subset L$ is a simple closed curve in $\mathbb{R}^{2}$.

Let $D_{1}$ be the image $f(S^{3})\subset\mathbb{R}^{2}$ and $N(L)$ a sufficiently small closed neighborhood of $L$. By removing the interior of $N(L)$ from $S^{3}$, we obtain the exterior $E(L)$, whose image $f(E(L))$ is a disk $D_{2}$ satisfying $D_{2}\subset D_{1}$. Note that $f(\partial E(L))$ is the union $f(\partial N(K_{1}))\cup\cdots\cup f(\partial N(K_{l}))$, and the image $f(\partial N(K_{i}))$ is a simple closed curve for each $i$. Note that the preimage of a point on $f(\partial N(K_{i}))$ is a meridian of $N(K_{i})$ under $f$. Let $U_{i}$ be a solid torus for each $i\in\{1,\ldots,l\}$. We glue $U_{i}$ to $E(L)$ via integral Dehn surgery on $K_{i}\subset L$ to obtain a $3$-manifold $M$. We remark that a longitude of $U_{i}$ is identified with a longitude of $N(K_{i})$ under the integral Dehn surgery. Then, a smooth map $f_{0}:M\to S^{2}$ is obtained by connecting $f\mid_{E(L)}$ and the natural projections $h_{i}:U_{i}\to\mathbb{D}^{2}$ so that $h_{i}(\partial U_{i})=f(\partial N(K_{i}))$. This map $f_{0}$ has no cusp points, and its singular set consists of indefinite fold points only. Moreover, $f_{0}$ satisfies the global conditions (5) and (6). Consequently, we obtain a stable map $f_{0}:M\to S^{2}$ such that $f_{0}$ has no cusp points, no definite fold points, and no singular fibers of type $\mathrm{I\hskip-1.2ptI^{3}}$, and $|\mathrm{I\hskip-1.2ptI^{2}}(f_{0})|=2(l-|X|)$ holds.

∎

Example 3.1*.*

We consider the minimally twisted five-chain link, $L10_{n}113$, in Figure 16. The link is well known in the study of exceptional Dehn surgery. See [11], for example. The link $L$ is represented by a $5$-string braid with the braid word $\sigma_{1}^{2}\sigma_{2}^{-2}\sigma_{4}^{-2}\sigma_{3}^{-1}\sigma_{2}\sigma_{1}^{-2}\sigma_{2}^{-1}\sigma_{4}^{2}\sigma_{3}$. Let $M$ be a closed orientable $3$-manifold obtained from $S^{3}$ by integral Dehn surgery on $L$. Then, a stable map $f:M\to S^{2}$ with $S_{0}(f)=\emptyset$ is obtained by the construction given in the proof of Theorem 1.3. In Figure 16, $f(S_{1}(f))$ is shown by immersed curves on $S^{2}$. This $f$ has no singular fibers of type $\mathrm{I\hskip-1.2ptI^{3}}$. Only some of the singular fibers are shown in the figure.

Acknowledgements

The author would like to thank his supervisor, Kazuhiro Ichihara, for his continuous guidance, invaluable advice, and warm encouragement. He also thanks Takahiro Yamamoto for usefull comments.