NEW ZEALAND JOURNAL OF MATHEMATICS Volume 30 (2001), 197-210
IMMERSED SURFACES AND THEIR LIFTS
T s u k a s a Y a s h i r o
(Received January 2000)
Abstract. In this paper we study immersed surfaces in 4-space and their pro jections in 3-space. W e use local deformations of immersed surfaces in 3—space to describe immersed surfaces in 4-space. W e construct an immersed lift into 4-space over an immersion in 3-space in which the double points are projected into neighbourhoods of the triple points of the immersed surfaces. This gives a construction of a regular homotopy from a knotted surface to a trivially knotted surface using immersed lifts. Then we define an isotopy invariant for knotted surfaces using a regular homotopy between knotted surfaces.
1. Introduction We assume that all manifolds are smooth. A map / : M m —> N n (m < n) is called an immersion if dfx has full rank for each x G M m. A knot is an embedded circle in R3. If we project a knot into R2, then we have an immersed circle after modifying the knot by an isotopy. Conversely, if an immersed circle with transversal crossings (see Section 2.1 for the definition) is given, then there is a knot such that the projection which coincides the immersed circle. This is not true in general for immersed surfaces in R3. Some immersed surfaces in R 3 are not projections of embedded surfaces in R4. This applies even for S2 [3] [1]. An embedded orientable surface in R 4 is called a knotted surface. Let f i , f 2 ' F 2 —+ R4 be embedded surfaces. f\ is equivalent to f 2 if there is diffeomorphism ip R4 —> R4 such that ip o f\(F2) = f 2(F 2). Note that this definition does not require ipo f x = f 2 as maps. We denote an equivalent class of f\ by [fi\. A knotted surface is trivial if it bounds a three dimensional handle body in R 4 ([8 ]). For a knotted surface, there is an invariant called an ‘unknotting number’ defined as the least number of 1 -handles attached to obtain a trivial knot. We construct an immersion of an orientable surface into R4, which is projected onto an immersed surface in R3. The constructed immersed surface in R 4 has double points. We show that an embedded surface in R 4 is regularly homotopic to a trivially knotted surface. The early result is established by Hirsch [5]. In the proof we construct the regular homotopy which is obtained from a regular homotopy of surfaces in R3. We introduce an isotopy invariant of knotted surfaces in R4. We call the invariant a ‘crossing number’. This is a generalisation of the unknotting number of knots.
2000 AM S Mathematics Subject Classification: 57M99, 57R42. 198 TSUKASA YASHIRO
2. Immersions 2.1. Crossings. Let M m and N n be compact m-manifold and n-manifold. and let / : M m —> N n (m < n) be an immersion. An immersion may have a crossing set defined as
C (f) = {x€M”‘ |#r 1 ( / ( x ) ) > l } , where # denotes the cardinality of a set. Since / is an immersion and M m is compact, / - 1 (/(x )) is a discrete finite point set. C (f) has disjoint subsets
Ck(f) = {x 6 C (f) | # (r1 (/(*))) = fc} ,(* = 2,3,...)• For x G C k (f), f{x ) is called a k-tuple point. The set of fc-tuple points form disjoint open submanifolds in N n. Note that a O-dimensional open submanifold is a point. The dimension d of each component of these submanifolds is given by
d = n — k(n — m). ( 1 )
Let f { x i) = f i x 2 ) = y. The immersion / is transversal at x\ G M m if dfXl(TXlM m) ^ dfX2{TX2M m) in TyN n. We assume that all immersions are transver sal at every point of C (f) and dimensions of components of C (f) follow the for mula ( 1 ). Two immersions f,g : M m —> N n are regularly homotopic if there is a homotopy H : M m x I —> N n such that setting Ht(x) = H (x,t) for x G M m, then Ho = / , Hi = g and each Ht is an immersion for all t G I except finite number of t, where Ht has non-transversal crossing points. We denote this by / ~ r g. If they are homotopic, then we denote f — g.
2.2. Singularities of maps. Let / : M m —► N n (m < n) be a map from an m-manifold to an n-manifold. Let x G M m and y = f(x ) and let {U,
3. Constructing Immersed Surfaces
3.1. h—moves. Given two homotopic surfaces in a 3-manifold, then the homotopy between these immersed surfaces can be realised by a sequence of six types of local deformations called h-moves [6 ] [7]. The h -moves are shown in Figure 2. The deformations from the left to the right are denoted by type*+ and in the opposite direction by IMMERSED SURFACES AND THEIR LIFTS 199
F i g u r e 1. Branch points.
Type IV Type V T y p e V I
F i g u r e 2. h -moves, Type I to Type VI.
The following two theorems can be found in [ 6 ] [7]. Theorem 3.1. Let f,g : F 2 —> N 3 be two immersions from a compact surface into a 3-manifold. If f — g, then there is a sequence of h-moves realising this homotopy. Theorem 3.2. Let f,g:F 2—> N 3 be two immersions from a compact surface into a 3-manifold. Then there is a sequence of h-moves which consist of type I, II, III and VI that deform f{F2) to g(F2) if and only if f ~ r g. In fact N 3 = R , types of these h-moves are viewed as projections of local isotopies of an embedded surface in IR4. Roseman introduced seven types of local deformations of surfaces in IR 4 called Roseman moves. That is a generalisation of ‘Reidemeister moves’ for knots (see Figure 11). An isotopy of knotted surface in
IR4 is expressed by a sequence of seven types of Roseman moves. Projections of Roseman moves consist of type I, II, III, IV and VI h-moves and the following two moves. (R l) Separate a pipe into pipes with two branch points or its converse (see the upper row in Figure 3), and (R2) a branch point passes through a double curve or its converse (see the lower row in Figure 3). 2 0 0 TSUKASA YASHIRO
F i g u r e 3. Two of Roseman moves
Rem ark 3.3. Deformations in Figure 3 are viewed as projections of isotopy moves in R4. We already know that these deformations are also decomposed into six types of h-moves as a homotopy in R3 [6 ] [7]. For instance, the upper row in Figure 3 is given as shown in Figure 6 . However, this does not mean that the sequence of h-moves in Figure 6 is the exact projection of the isotopy used for the deformation in R4.
3.2. Bugs. In the following, we will discuss immersed surfaces in R3. A double segment in an immersed surface is a double curve terminated by triple points or branch points. Applying a type IV+ move on a disk, we obtain a pair of branch points and a double segment terminated by these branch points. We call the singular disk with a pair of branch points a bug. Let F 2 be an orientable closed surface and g : F 2 —> R3 be an embedding. Let 7 : S1 —> F 2 be an oriented immersion. Choose a point p of the immersed circle g o 7 (S'1) and suppose that p is not a crossing point of g o 7 (S1). Put a bug in a neighbourhood U(p) of p in F2 where the double segment of the bug is on the arc g o 7 (S'1) D U(p). Producing the bug changes / to a map with branch points. We can lengthen the bug along g o 7 on g{F2) (Figure 4).
F i g u r e 4. Lengthening a bug. IMMERSED SURFACES AND THEIR LIFTS 2 01
We call one branch point of a bug the head and the other the tail. When the head passes a part of the bug the deformation produces triple points in a neighbourhood of the crossing point of g o 7 (S'1) (see Figure 5) [6 ] [16].
F ig u r e 5. Passing through a double curve.
Finally, the bug head meets its tail. We can eliminate the pair of branch points to obtain an immersed surface. The elimination is done using type VI and type IV- h-moves as shown in Figure 6 [6 ] [7]. Note that these deformations are homotopy deformations but it is not a regular homotopy.
F ig u r e 6 . Eliminating a pair of branch points.
3.3. Bug constructions. As we have seen we can construct an immersed surface from an embedding g : F 2 —> R 3 and an immersed circle 7 : S'1 —* F 2. The procedure of the construction is the following.
(i) Create a bug on g 0 7 ,
(ii) lengthen the bug along g 0 7 , and (iii) cancel the pair of branch points when they meet. We call this a bug construction. We denote the constructed immersion by g7. This immersion is well defined up to regular homotopy [16]. 2 0 2 TSUKASA YASHIRO
Remark 3.4. Hass and Hughes also showed that the 7 induces an immersed sur face [4]. They use cut and paste technique.
4. Lifts of Immersions In the following, F 2 denotes an orientable closed surface. Let X , Y and Z be topological spaces and let f : X —>Y, f : X —> Z and p : Z —» Y be maps. Then / is called a lift over f via p if / = p o f. Let p : R4 —> R3 be the projection defined by p(xq,x\,X2,X3) = (xi,X2,xs). We will construct an immersed lift in R4 over an immersion / : F 2 —> R3 via p.
4.1. Constructing immersed lifts. Let / : F 2 —> R3 be an immersion. The crossing set C(f) consists of finite number of immersed circles c* : S1 —■> F2 (i = 1 ,2 ,... ,k). We denote E(ui) the total space of the normal bundle Vi of Ci over S1. Each ct can be extended to an immersion ci : E(ui) —*■ F 2. Since F 2 is orientable, the total space E(ui) is diffeomorphic to S 1 x (—1,1). We may assume that the crossing set of {c*} is a union of squares around a crossing point of cl. Let
x = 01 ((li (em ) ) \ (u C@)J j • <2)
Then X is a union of disks for which the image f(X) contains only double curves. A tubular neighbourhood of each double curve in R3 has a cross section, which contains two arcs intersecting at their midpoints shown in Figure 7. Assume that we view f(F2) as a subset of R3 x {0} C M4. We add crossing data to the crossing to give a height function so that the lift of X into R4 is an embedding. We introduce
F ig u r e 7. A cross section of the neighbourhood of a double curve and its lift. some diagrams which describe a neighbourhood of a triple point of f(F2) in R3. Let J3 = (—1,1) x (—1 , 1 ) x (—1,1) be a cube centred at the origin 0. There are three open rectangles in J 3 such as (—1,1) x (— 1 , 1 ) x {0}, {0} x (—1,1 ) x (—1,1) and (—1,1) x {0} x (—1,1). These three rectangles intersect each other and make six double curves and one triple point. Let T be the union of the three rectangles (see Figure 8 ). Choose a vector v = (1,1,1) and a plane P being perpendicular to v. If the distance between the origin and P is less than l/\/3, then P D T gives diagrams consisting of three segments as P moves along the line directed by v. In Figure 8 the upper row shows only segments of P fl T on T. IMMERSED SURFACES AND THEIR LIFTS 2 0 3
F ig u r e 8 . P (IT consists of three edges.
Let p be a triple point of f(F2) C R3. Choose a neighbourhood U(p) of p in
R 3 so that U(p) fl f(F2) has the same structure as T. Thus we can describe the relation between double curves around p as T fl P (see Figure 8 ). In Figure 9, (a) shows parts of T fl P in P. And (b), (c) and (d) show the possi bilities of lifts of the diagram (a). In each diagram each crossing point corresponds to a double point. The arrow indicate the movement of P in the neighbourhood U(p). The case (d) requires at least two crossing changes. We assume that for each
(c) (d)
F ig u r e 9 case of (b), (c) and (d), the number of crossing points in R 4 can be minimised. Remind that a loop is a simple closed curve based at a point.
Lemma 4.1. Let p be a triple point of f(F2) c R3. If there is a loop in C(f) based at a point of f ~ 1(p), then the case (d) does not occur.
P roof. Let / - 1 (p) = {go>Qi,Q2} and let £ be the loop on F 2 based at qo. Since f{£) is a double curve, there is an edge e\2 such that q\ and q2 are end points of an edge ei 2 in C(f) so that f(£) = f(e 12). 2 0 4 TSUKASA YASHIRO
Let U(p) be a neighbourhood of p in R3. Then U(p) fl f{F2) is homeomorphic to T so f~1(U(p) n f(F2)) has three disks i?0, Ri and R2 containing q0, qi and q2 respectively. Recall that X is defined by the formula (2). It is obvious that the restriction map / | X has an embedded lift into R4. The lift can be extended to F2 as an immersion / : F2 —> M4. Using the same notation in the formula (2), we assume that C(f) C Uk=1C(ci). Thus Cl(ei2 \ (R\ U R2)) and Cl(£ \ Ro) are embedded in R4 by / . Since t is a loop, we assume that / (Cl(ei2 \ {Ri U R2))) is mapped above /(C l(^\ Ro)) with respect to the first coordinate. We can choose a plane P near p in R3. There are three arcs in T fl P. We can assume that T fl P is also lifted into R4 as an embedded three arcs. However, one of these arcs connecting i is below or above both two other arcs. This means that the case (d) does not occur. □
We can construct an immersion / over / using the above liftings. Over a triple point the lift has at most 2 crossing points in R4. Hence the following lemma holds. Lemma 4.2. An immersion f : F 2 —► R3 has an immersed lift f : F 2 —► R4; which has double points projecting to near triple points of f(F2). For a fixed embedding g : F 2 —* R3, we have the following [4] [16]. Theorem 4.3. For an immersion f : F 2 —> R3, there is an immersion 7 -.S1 ^ F 2 such that g7 ~ r / .
Assume that 7 has transversal self-intersections. From the bug construction gy has triple points each of which joins looped double curves. By Lemma 4.2, each double point of the lift into R4 is projected into a neighbourhood of a triple point of g1 in R3.
Lemma 4.4. g7 has an immersed lift p7 into R4 such that the crossing points of gy projected onto triple points of g7.
Proof. We can construct g 7 by Lemma 4.2. From the bug construction, there is a point x G C^g^) such that x has a loop in C(g1) C F 2. Thus from Lemma 4.1 for each x £ Cs(gy), the lift of a neighbourhood of gy{x) into R4 contains at most one double point. Then we move the double point over a triple point by a regular homotopy. □
The regular homotopy class of gy is determined by a homology class of H 1 (F 2; Z 2) represented by the immersed circle 7 on F 2 [4] [16]. This implies that the homology class [ 7 ] is also represented by a union of disjoint embedded circles on F2. Let <5 be such a set of circles. We can deform an immersion g1 to gs for a set 5 using h-moves [16]. This modification deforms each intersection of pipes of g7 to split pipes (see Figure 10).
Lemma 4.5. g$ has an embedded lift g$ into R4 such that g$ is trivial in R4. Proof. The immersion has pipes along disjoint circles on g{F2). Applying Rose man moves (RI) to each pipe, we can obtain a pair of branch points. Thus we obtain bugs on g(F2). Bugs are eliminated by Roseman moves (or type IV- h-move). □ IMMERSED SURFACES AND THEIR LIFTS 2 0 5
F i g u r e
Remark 4.6. Immersed surfaces in R 4 do not always project onto an immersed surface in R3. If the projection is not an immersed surface, then we can assume that the projection has some branch points.
We have the following.
Theorem 4.7. Let f : F 2 —> R 3 be an immersion and let g : F 2 —> R3 be an embedding. Suppose that / : F 2 —> R4 and g : F 2 —> R4 are embedded lifts over f and g respectively. Then f is regularly homotopic to g covering a homotopy from f to g.
Proof. There is a set of disjoint embedded circles 6 on F 2 such that / is regularly homotopic to gs in R3. Therefore, there is a regular homotopy H : F 2 x I —* R3 such that H0{x) = f(x) and H\(x) = gs- From Theorem 3.2 there is a se quence of type I, II, III and VI h-moves to realise the regular homotopy H. Let k be the number of these h-moves in the sequence. There are subintervals I1 = [ 0 = to,ti],... ,/fc = [tk-i,tk = 1] such that H is decomposed into k regular homotopies {Ht;(t e [if, ' H(x,2t) if 0 < f < 1/2 H*G(x,t) = G(x,2t-l) if 1 / 2 < t < 1. gives the required regular homotopy. □ 2 0 6 TSUKASA YASHIRO 5. Crossing Numbers In knot theory the unknotting number is a knot invariant. It is defined as the least number of crossing changes necessary to obtain a trivial knot. Viewing the crossing changes as regular homotopy deformations, the unknotting number is generalised as the least number of crossing points of the regular homotopy track on 5 1. The track is an immersion from an annulus into R4. We apply this idea to a knotted surface in R 4 to obtain a knotted surface invariant. There is an unknotting number for knotted surfaces defined as the least number of attaching 1-handles to obtain a trivially knotted surfaces (see [ 8 ]). While we introduce an isotopy invariant defined by the least number of crossing components in the track obtained from a regular homotopy from a knotted surface to a trivially knotted surface. 5.1. The unknotting number. It is known that an unknotting number of a knotted surface / : F2 —* R 4 is defined as the least number of 1 -handles attaching to [/] to obtain a trivially knotted surface [ 8 ]. Assume that [/] is represented by a surface F2 c R . Let B\,... , B 3 be a set of closed 3-balls. Then we attach Bf to F2 as a 1-handle for each i = 1,... , u. Then we obtain a surface hx(F2-,Bf,... , B 3) = F 2\j(dBfu■ • -U<9.B3 )\.F2 n(£?3 lJ- • -Ui?3) (see [8 ]). If the surface hl (F 2-, Bf,... , J53) is trivial, then the least number of such 1-handles is called an unknotting number of [/]. This is denoted by «([/]). Remark 5.1. It is obvious that the trivial surface obtained from [/] by attaching 1 -handles has different genus. The modification of a knotted surface with attaching 1-handles can be viewed as attaching 1-handles to I x F 2 C R x R4 to obtain a cobordism between F 2 C R4 and hl(F2-Bf,... ,S3) c R4. 5.2. C(H). The unknotting number in knot theory is defined as the least number of crossing changes to obtain a trivial knot. Viewing the crossing changes as a regular homo topy H : S1 x I —> R3, then we obtain a regular homotopy track H : I x S1 —* I x R3. Therefore, the least number of crossing changes to obtain a trivial knot is the least number of crossing points in the image of H. We generalise this idea to knotted surfaces in R4. Let H be the regular homotopy from a knotted surface / to a trivially knotted surface g, H : F2 x I -»• R4, such that setting Ht(x) — H(x,t), H0(x) = f(x) and Hi(x) = g(x) for all x € F2. This homotopy induces a track, H : I x F2 -> I x R4 c R5 defined by H(t,x) = (t,Ht(x)). H is an immersion from a product 3-manifold I x F2 into R5. Thus since the formula (1) and I x F2 is compact, we may assume that C(H) consists of disjoint circles or arcs in I x F2. We can eliminate these arcs because the ends of the track are embedded surfaces. IMMERSED SURFACES AND THEIR LIFTS 2 0 7 We define an isotopy invariant for knotted surfaces in R4. The least number of components of C(H) for all H will be called a crossing number. We denote this by y(D- Theorem 5.2. The crossing number y ( f) is an isotopy invariant. P roof. We fix an orientable surface F 2. Let fi and f 2 be knotted surface in R4. Let g be a trivially knotted surface in R4. Then there is a regular homotopy class of a track H : I x F 2 —> I x R4 such that H(0,x) = (0, fi{x)), H(l,x) = (l,g(:r)) and C(H) has the minimum number of components. Assume that f2 is isotopic to f\. This isotopy gives a track G having no crossings. Thus the composite track G(2t,x) if 0 < i < 1/2 G*H{t,x) = H(2t-l,x) if 1/2 < t < 1 gives the same number of the components in C(H). □ 6 . Experimental Results The above arguments suggest us to find the unknotting number of knots is to find a regular homotopy which has the minimal crossings. In this section we apply this idea to knots and we define a number of crossing changes for a particular knot diagram. Then we compare the unknotting number of the knot and our number of it. 6.1. Unknotting numbers of knots. Knots are modified by an isotopy. The isotopy is decomposed into three types of Reidemeister moves (see Figure 11). -Q _-JVA— A -X-“X Type 1 Type 2 Type 3 F i g u r e 11. Reidemeister moves We can modify a knot K : S1 —> R3 to a trivial knot using regular homotopy in R3. We may define a procedure of the deformation of a knot diagram to obtain a trivial knot as follows. (1) Find a loop in the diagram of K, which makes the minimal crossing changes if it is shrunk, (2 ) shrink the loop until it becomes a trivial loop, (3) apply the Reidemeister move type 1 to eliminate the loop and (4) continue from (1) to (3) until K becomes trivial. 2 0 8 TSUKASA YASHIRO F ig u r e 12 . Deformations using three loops of ^9,45. This procedure gives a regular homotopy of a circle in R3. Thus the regular homotopy gives an immersed track in R4. Note that for one-dimensional case, the track forms an immersed annulus. We define the number y(K) of crossing points in the immersed annulus by y(K) = where e* is the number of crossing changes in the stage (1 ). For example, -Kg, 45 is deformed using three loops (see Figure 12). This sequence of modifications in R2 gives an immersed annulus as a track in R4. It has three crossing points. Thus y{K\9,4 5 ) = 3. On the other hand, the unknotting number u (K q ^ ) = 1. This is FIG U R E 13 . The crossing change on .# 9 ,4 5 obtained by changing one crossing (see Figure 13). The shaded circle shows the crossing change. An immersed annulus obtained from this deformation has only one crossing point in R4. This deformation can be viewed as a deformation in R3 so that the deformation forms an immersed annulus in R4 with one crossing point. The table (next page) shows y = y{K), u = u(K) and y — u for some knots dia grams based on a table of Kauffman’s text book [12]. The difference between y(K) and u(K) comes from the difference between tracks obtained by the deformations. IMMERSED SURFACES AND THEIR LIFTS 2 0 9 K y u y -u K V u y-u K y u y-u 0 0 0 Ks,\5 2 2 0 Kg,23 2 2 0 K3, i l 1 0 K8,16 3 2 1 Kg,24 3 1 2 K4, i l 1 0 K8,l7 2 1 1 Kg,25 2 2 0 K*j,i 2 2 0 K8,18 2 2 0 Kg,26 3 1 2 K 5 ,2 1 1 0 K s,19 3 3 0 Kg,27 3 1 2 K q,\ 1 1 0 K8,20 1 1 0 Kg,28 3 1 2 K6,2 2 1 1 Ks,21 1 1 0 Kg,29 2 1 1 K6,3 2 1 1 Kq,i 4 4 0 Kg,30 3 1 2 k7<1 3 3 0 Kg,2 1 1 0 Kg, 31 3 2 1 ? K7,2 1 1 0 Kg,3 3 3 0 Kg,32 3 ? k7,3 2 2 0 Kg,4 2 2 0 Kg,33 3 1 2 K7,4 2 2 0 Kg,5 2 2 0 Kg,34 2 1 1 K 7,5 2 2 0 Kg,e 3 3 0 Kg,35 3 ? ? K7,q 2 1 1 Kg,7 2 2 0 Kg,36 2 2 0 K 7,7 2 1 1 Kg,s 2 2 0 K9,37 2 2 0 Xs.l 1 1 0 Kg,g 3 3 0 Kg,38 2 ? ? Ks,2 3 2 1 Kg, io 3 ?? Kg,39 3 1 2 K8,3 2 2 0 Kg,ii 3 2 1 Kg,40 2 2 0 K&,4 2 2 0 Kg,12 2 1 1 Kg,4\ 2 2 0 K8,5 3 2 1 Kg, 13 3 ? ? Kg,42 1 1 0 Ks,6 2 2 0 -^9,14 2 1 1 Kg,43 3 2 1 Ks,7 3 1 2 Kg, 2 2 0 Kg,44 2 1 1 $ 00 2 2 0 Kg,ie 3 3 0 Kg,45 3 1 2 Kg,9 3 1 2 Kg,\7 3 2 1 Kg,46 2 2 0 Kg,xo 3 ? ? Kg,is 2 2 0 Kg, 47 2 2 0 Kg, ii 2 1 1 Kg, 19 2 1 1 Kg, 48 2 2 0 ?? 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Nagase, On regular homotopy of a closed orientable surface into a 3-manifold, Proc. Fac. Sci. Tokai Univ. 19 (1984), 35-41. 15. U. Pinkall, Regular homotopy classes of immersed surfaces, Topology, 24 (1985), 421-434. 16. T. Yashiro, Constructing immersions from three manifolds into four dimen sional space, Thesis, The University of Auckland. Tsukasa Yashiro Department of Mathematics The University of Auckland Private Bag 92019 Auckland NEW ZEALAND yashiro@math .auckland. ac. nz