数理解析研究所講究録
第 1764 巻 2011 年 165-173

165

ON IMMERSED ORIENTED SURFACES AND THEIR
PLANE PROJECTIONS
愛知教育大学 )MINORU YAMAMOTO ( 山本 稔 ，DEPARTMENT OF MATHEMATICS, AICHI UNIVERSITY OF EDUCATION

1. INTRODUCTION

The author has been supported by Grant-in-Aid for Young Scientists (B) JSPS (No.
21740058).

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2. STABLE MAP
In this section, we give the definition and see properties of a stable map.
Let $f$ :be a smooth map of a closed connected surface $M$, which isinto the plane. We denote the set of such maps by$f$-topology. A smooth map is said to be aequipped with the Whitney
,map
if in
there exists an open neighborhood of suchstableright-left equivalent to $f$ , i.e., there exist twothat for anyis$MarrowM$such that the diagram
diffeomorphisms :and$Marrow \mathbb{R}^{2}$$C^{\infty}(M, \mathbb{R}^{2})$$C^{\infty}$$U$$C^{\infty}(M, \mathbb{R}^{2})$$\tilde{g}\in U,\tilde{g}$$\tilde{f}$$C^{\infty}$$\varphi:\mathbb{R}^{2}arrow \mathbb{R}^{2}$$\Phi$$Marrow^{\Phi}M$$\downarrow\tilde{g}$$\overline{f}\downarrow$$\mathbb{R}^{2}arrow \mathbb{R}^{2}$$\varphi$

is commutative.
, we denote by $s(f)$ the set of the points
For a smooth map
in $M$ where the rank of the differential of is strictly less than two. We
say thatis a singular set of $f$ andis an apparent$f$.
contour ofThe following characterizations of stable maps are well-known (see [2, 8],
for example).
$f;Marrow \mathbb{R}^{2}$$\tilde{f}$$f(S(f))\subset \mathbb{R}^{2}$$S(\tilde{f})\subset M$

Proposition 2.1. A smooth mapof a closed surface $M$ is astable map if and only if the following conditions are satisfied.
(i) For every $q\in M$ , there $ex:ist$ local coordinates $(x, y)$ and (X, Y)
respectively such that one of the folaround $q\in M$ andlowing holds:$(a)(X\circ\tilde{f}, Yo\tilde{f})=(x, y)$ (: regular point),$(b)(Xo\tilde{f}, Yo\tilde{f})=(x, y^{2})$ (: fold point),$(c)(Xo\tilde{f}, Yo\tilde{f})=(x, y^{3}-xy)$ ( : cusp point).
(ii) If $q\in M$ is a cusp point, then $\tilde{f}^{-1}(f(q))\cap s(f)=\{q\}$ .
(iii) The map (points}) is an immersion with normal crossings.
$f;Marrow \mathbb{R}^{2}$$f(q)\in \mathbb{R}^{2}$

$q$

$q$

$q$

$f|$

$S(\tilde{f})\backslash \{cusp$

Note that $s(f)$ is a compact l-dimensional submanifold of $M$ and thenumber of cusp points is finite. Let $U\subset M$ be a tubular neighborhood of. Then the restriction of on the closure clis an immersion.
By the apparent contour of $f,$is naturally stratified into 2-, 1- and 0dimensional strata. The union of l-and 0-dimensional strata forms.
canstratum,
weanorientation
On each l-dimensional
as follows.
define. Let be a connected component
We fix the canonical orientation on, which. We associate to a non-negative integerofis the number of points in the fiber of over any point of . Every 1dimensional stratum is adjacent to exactly two connected components of-values, we. Since these two components have distinct
can orient each l-dimensional stratum inso that the region withcusp points}) is an
the larger-value is on its left. Since (immersion,{cusp points} is also oriented.
$f$$S(\tilde{f})$$(M\backslash U)$$\mathbb{R}^{2}$$\tilde{f}(S(f))$$\mathbb{R}^{2}$$\mathbb{R}^{2}\backslash \tilde{f}(S(\tilde{f}))$$\Omega$$\Omega$$n_{\overline{f}}(\Omega)$$\Omega$$\tilde{f}$$\mathbb{R}^{2}\backslash \tilde{f}(S(\tilde{f}))$$n_{\overline{f}}(\Omega)$$f(S(\tilde{f}))$$n_{f}(\Omega)$$S(\tilde{f})\backslash$$f|$$S(\tilde{f})\backslash \{$167
Suppose that $M$ is an oriented closed surface andis oriented plane.
Let be a cusp point of a stable map. For a sufficiently smallneighborhood, whereof $f(q)$ , the map $f|v$ : $Varrow U$ has degree$f^{-1}(u)$is the component ofcontaining . We call is a positive (resp.
negative) cusp if the local degree of $f$ at equals $+1$ (resp. $-1$ ).
$\mathbb{R}^{2}$$f;Marrow \mathbb{R}^{2}$

$q$

$U$

$V$

$\pm 1$

$q$

$q$

$q$

3. IMMERSION LlFT
In the following, we assume that $M,$
andHaefliger theorem is restated as follows.
$\mathbb{R}^{3}$$\mathbb{R}^{2}$are oriented. In this case,$s$

Theorem 3.1 (Haefliger [3]). A stable map
lift if and only if each connected component ofpoints.

has an immersion
has even number of cusp

$f;Marrow \mathbb{R}^{2}$$S(\tilde{f})$Letbe a stable map which has an immersion lift. On eachconnected component of fold points{cusp points}, we can put a sign$+1$ or-l which satisfies the following rule.
Let and $C’$ be two connected components which adjacent to thesame cusp point. Then and C’ have the opposite signs.
If a sign of is $+1$ (resp. $-1$ ), we call a positive (resp. negative) fold anda sign can be put on each image of fold component. Such a stable map iscalled a signed stable map.
Letbe a signed stable map anda tubular neighborhood. Since $M$ is oriented,
ofis divided into two regions $U+$ and-where $f|U+$ (resp. $f|U_{-}$ ) is an orientation preserving (resp. reversing):immersion. We construct an immersion lift
over $f|u$ which
satisfies the following.
(1) If is a positive fold, is defined as Figure 1 (a).
(2) If is a negative fold, is defined as Figure 1 (b).
(3) If is a positive cusp and negative fold comes in for the orientation
ofis defined as Figure 2(a).
(4) If is a positive cusp and positive fold comes in for the orientation
ofis defined as Figure 2(b).
(5) If is a negative cusp and negative fold comes in for the orientationof $s(f),$ is defined as Figure 2(c).
(6) If is a negative cusp and positive fold comes in for the orientationof $s(f),$ is defined as Figure 2(d).
$f:Marrow \mathbb{R}^{2}$$S(\tilde{f})\backslash$$C$$\bullet$

$C$

$C$

$C$

$\tilde{f}$$f:Marrow \mathbb{R}^{2}$$U$$U\backslash s(f)$$S(\tilde{f})$$U$$f_{U}$

$C$

$f$

$C$

$f$

$Uarrow \mathbb{R}^{3}$

$q$

$q$

$S(\tilde{f}),$

$f$

$q$

$q$

$S(\tilde{f}),$

$f$

$q$

$q$

$f$

$q$

$q$

$f$

Definition 3.2. Leta tubular
be a signed stable map andneighborhood of. If an immersion :satisfies the above rules(1) $-(6)$ on $f|U$ , we call an immersion lift over the signed stable map $f$ .
$f;Marrow \mathbb{R}^{2}$$U$$S(\tilde{f})$$f$$Marrow \mathbb{R}^{3}$

$f$

4.

-REGULAR HOMOTOPY$f$

In this section, we state that $f$-regular homotopy classes can be determined.

Theorem 4.1. If $f$ and $g:Marrow \mathbb{R}^{3}$ are immersion lifts over the signed
stable mapthen and are $f$ -regularly homotopic.
$f:Marrow \mathbb{R}^{2_{f}}$

$f$

$g$

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$\downarrow\pi$$\downarrow\pi$

(b)

(a)
FIGURE 1. Immersion lifts if (a)is a negative fold.

$C$

is a positive fold, (b)$C$

Corollary 4.2. Let
be a stable map which has an immersion$f$signed.)(NotethatThenotnumber of $f$ -regular homotopy classes
islift.
, whereis.
is the number of connected components of$f:Marrow \mathbb{R}^{2}$$2\# S(\overline{f})$$S(\tilde{f})$$\# S(\tilde{f})$We have a following example which is related to Theorem 4.1.
Example 4.3. Let
be an oriented torus and and $m$ longitude and, respectively. Let $f$ and:meridian of
be signed stablemaps which satisfy the following properties. They do not have cusp points,, both signs are the same,andare plane$f|m$curves whose rotation numbers equa12 (or-2),
is a simple closed plane
curve. See Figure 3. By the theorem of Pinkall [6], immersion lifts $f$ andover $f$ and respectively are not regularly homotopic.
$T^{2}$$l$$T^{2}$$\tilde{g}$$f(S(\tilde{f}))=\tilde{g}(S(\tilde{g}))$$T^{2}arrow \mathbb{R}^{2}$$f|\iota=\tilde{g}|l$$g:T^{2}arrow \mathbb{R}^{3}$$\tilde{g}|m$$\tilde{g}$Theorem 3.1 and Example 4.3 mean that if $M\neq S^{2}$ , an apparent contourwith sign does not determine a regular homotopy class. We need informationof immersion $f|(M\backslash s(f))$ .

5. REGULAR HOMOTOPY LIFT OVER A GENERIC HOMOTOPYLet $f$ and :be stable maps and :a homotopy$f$between and . If satisfies the following conditions, we call a generic
homotopy between $f$ and (see [5]).
(1) There is a finite set of parameter values $0<t_{1}<\cdots<t_{n}<1$(possibly empty) in $(0,1)$ .
(2) For any:
is a stablemap.
(3) For eachand a sufficiently small positive value , the moves ofapparent contours ofare classified into
lips (type ), beaks (type ), swallowtail (type ), cusp-fold (type$C)$ , self-tangency (type $K$ ) or triple point (type).
$\tilde{g}$$Marrow \mathbb{R}^{2}$$\tilde{F}$$M\cross[0,1]arrow \mathbb{R}^{2}$$\tilde{F}$$\tilde{F}$$\tilde{g}$$\tilde{g}$$t\in(0,1)\backslash \{t_{1}, \ldots, t_{n}\},\tilde{F}|M\cross\{t\}$$M\cross\{t\}arrow \mathbb{R}^{2}$$t_{i}$$\epsilon$$\tilde{F}|M\cross\{t\}(t\in(t_{i}-\epsilon, t_{i}+\epsilon))$

$L$

$B$

$S$

$T$

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$\downarrow\pi$$\downarrow\pi$

(a)

(b)

$\downarrow\pi$$\downarrow\pi$

(c)

(d)

FIGURE 2. Immersion lifts if (a) is a positive cusp andnegative fold comes in, (b) is a positive cusp and positive
fold comes in, (c) is a negative cusp and negative fold comesin, (b) is a negative cusp and positive fold comes in.
$q$

$q$

$q$

$q$

We call each is a bifurcation point on a generic homotopy .
Let:
be a generic homotopy between signed stable$f$maps and and let and immersion lifts over $f$ and , respectively. If
there exists a regular homotopy $F:M\cross[0,1]arrow \mathbb{R}^{3}$ between and suchthat, we call $F$ a regular homotopy lift over .
$\tilde{F}$$t_{i}$$\tilde{F}$$M\cross[0,1]arrow \mathbb{R}^{2}$$f$$\tilde{g}$$g$$\tilde{g}$$f$$\pi\circ F=\tilde{F}$$g$$\tilde{F}$

Theorem 5.1. Let $f$ anda generic homotopy :$\tilde{F}$$\tilde{g}$:$Marrow \mathbb{R}^{2}$$M\cross[0,1]arrow \mathbb{R}^{2}$

be signed stable maps. If there exists
between $f$ and which preserves sign$\tilde{g}$170$\downarrow\pi$$\downarrow\pi$which satisfy
FIGURE 3. Two stable maps and :and both apparent contours havethat
positive signs. But their immersion lifts andare not regularly homotopic.
$f$$T^{2}arrow \mathbb{R}^{2}$$\tilde{g}$$\tilde{f}(S(\tilde{f}))=\tilde{g}(S(\tilde{g}))$$f$convention as depicted in Figures 4 and 5, thenlift$F:M\cross[0,1]arrow \mathbb{R}^{3}$.

$\tilde{F}$$g:T^{2}arrow \mathbb{R}^{3}$

has a regular homotopy

As an application of Theorem 5.1, we have the following example.

are signed stable maps such thatis a simple closed curve and the) is $+1$ (resp. $-1$ ). Then there is a generic homotopy
between and which has a regular homotopy lift

Example 5.2. If $f$ and$f(s^{2})=\tilde{g}(S^{2})=D^{2},$sign of:$F$ :$\tilde{F}$$S(\tilde{f})$(resp.

$\tilde{g}$:$S^{2}arrow \mathbb{R}^{2}$$f(S(\tilde{f}))=\tilde{g}(S(\tilde{g}))$$S(\tilde{g})$$S^{2}\cross[0,1]arrow \mathbb{R}^{2}$$f$$\tilde{g}$, the regular. See Figure 6. By the definitions ofhomotopy lift over corresponds to an eversion of the embedded sphere.
Our eversion in Example 5.2 is almost same as the eversion given by
Flrancis [1]. But in his picture, self intersections of immersed spheres were not$f,\tilde{g}$$S^{2}\cross[0,1]arrow \mathbb{R}^{3}$$F$$\tilde{F}$drawn. Professor Mikami Hirasawa and the author draw a regular homotopy
over the generic homotopy of Figure 6, precisely. So, we can follow how selfintersections move during our sphere eversion. Our eversion will appear in
their preparing paper,REFERENCES[1] G. K. Francis, A topological picturebook, Springer, New York, 2007.
[2] M. Golubitsky and V. Guillemin, Stable mappings and their singularities, Graduate Texts in Mathematics, Vol. 14, Springer-Verlag, New York, Heidelberg,

1973.

[3] A. Haefliger, Quelques remarques sur les applications differentiables d’une surface dans le plan, Ann. Inst. Fourier Grenoble 10 (1960), 47-60.

171$rightarrow^{L}$$rightarrow^{B}$$rightarrow^{S}$

FIGURE 4. Bifurcations of type $L,$ $B,$ andwhich haveregular homotopy lifts. Here,and $\beta=\pm 1$ and andvary independently.
$S$$\alpha=\pm 1$$C$$\alpha$$\beta$[4] I. James and E. Thomas, Note on the classification of cross-sections, Topology
4 (1966) 351-359.
[5] T. Ohmoto and F. Aicardi, First order local invariants of apparent contours,Topology 45 (2006), 27-45.
[6] U. Pinkall, Regular homotopy classes of immersed surfa ces, Topology 24 (1985),

421-434.

[7] S. Smale, A classification of immersions of the two-sphere, Trans. Amer, Math.
Soc. 90 (1958), 281-290.

172$rightarrow^{K_{1}}$$rightarrow^{K_{2}}$$rightarrow^{T_{1}}1$$rightarrow^{T_{2}}$FIGURE 5. Bifurcations of type $K$ and which have regular$\beta=\pm 1$ andandhomotopy lifts. Here,independently.
and vary$T$$\alpha=\pm 1,$$\gamma=\pm 1$$\alpha,$$\beta$$\gamma$[8] H. Whitney, On singularities of mappings of euclidean spaces. I. Mappings ofthe plane into the plane, Ann. of Math. (2) 62 (1955), 374-410.

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FIGURE 6. A sequence of apparent contours of a generic homotopy between $f$ andwhich has a regular homotopy lift. This regular homotopy corresponds to a sphereeversion.
$\tilde{g}:S^{2}arrow \mathbb{R}^{2}$