On pseudo-horizontal surfaces and \mathbb{Z}_2-Thurston norms in small Seifert 3-manifolds
aa r X i v : . [ m a t h . G T ] F e b ON PSEUDO-HORIZONTAL SURFACES AND Z -THURSTONNORMS IN SMALL SEIFERT -MANIFOLDS XIAOMING DU
Abstract.
We give a criterion for the existence for pseudo-horizontal surfacesin small Seifert fibered manifolds. We calculate the genuses for such surfacesand detect their Z -homology classes. Using such pseudo-horizontal surfaces,we can determine the Z -Thurston norm for every Z -homology classes in smallSeifert manifolds. We find several families of examples that in the same Z -homology class the genus of a pseudo-horizontal surface is less than the genus ofthe pseudo-vertical surface. Hence the pseudo-vertical surfaces is not Z -taut. Introduction
Let M be a closed orientable, irreducible, connected 3-manifold. Let α bea non-zero element in H ( M ; Z ). Let S denote an closed embedded surfacerepresenting α . Such an S can be non-orientable. It can also be disconnected.We can define the Z -Thurston norm of α , i.e., k α k Z − T h = min [ S ]= α (X i max { , − χ ( S i ) } ) , where S varies all surface that represents α , S i ’s are the components of S , χ ( S i )is the Euler characteristic, and the summation is over all components. A lowerbound for the complexity of M is given by calculating the Z -Thurston norm ofevery non-zero element in H ( M ; Z ), see [5, 6]. The surface S representing α issaid to be Z -taut if no component of S is a sphere and χ ( S ) = −k α k Z − T h .When S is Z -taut, non-orientable, connected, and not a projective plane, the Z -Thurston norm of [ S ] = α is g ( S ) −
2, where g ( S ) is the genus of S , i.e.,the number of R P ’s in the connected sum decomposition of S . The genus ofsuch an S is minimal among all connected embedded non-orientable surfaces Mathematics Subject Classification.
Key words and phrases.
Seifert manifold, incompressible surface, Z -Thurston norm.The author would like to thank Hyam Rubinstein for valuable comments. The author alsothank Adam Levine, Zhongtao Wu, Jingling Yang for their helps on the calculation of thecorrection term. representing α . As mentioned in [5], S is geometric incompressible. Conversely,for a closed connected embedded non-orientable surface representing α , how dowe know whether it is incompressible? The only ‘easy’ way is to check if it isof minimal genus in its Z -homology class and check the Z -homology class doesnot come from a Z -homology class. For every non-zero element in H ( M ; Z ),we want to know the minimal genus of the representing connected embeddednon-orientable surfaces.From now on, for abbreviation, we use the word “one-sided surface” to referconnected embedded non-orientable surface in an orientable 3-manifold.In Seifert fibered manifolds, Frohman [2] proved that a geometric incompress-ible one-sided surface is isotopic to a pseudo-vertical surface or a pseudo-horizontalsurface. So in this case we only need to find all pseudo-vertical surfaces andpseudo-horizontal surfaces that represent a given non-zero element in H ( M ; Z )and compare their genuses. The embedded non-orientable connected surface withminimal genus is Z -taut and geometric incompressible.Pseudo-vertical surfaces have been studied by Frohman [2]. There is no litera-ture on pseudo-horizontal surfaces. We will study the pseudo-horizontal surfacesystematically, calculate their genus, and detect their Z -homology classes. Thegenus relies on a recursive function N (2 k, q ), see Section 2. With this in hand, wecan efficiently calculate the Z -Thurston norm for every non-zero Z -homologyclasses in small Seifert manifolds. The result of the calculation reveals that inmost cases the pseudo-vertical surfaces are Z -taut, i.e. we can use them to cal-culate the Z -Thurston norm. However, we find several families of small Seifertmanifolds. In each of these manifolds, there exists a pseudo-horizontal surfacewhose genus is less than the pseudo-vertical surface in the same Z -homologyclass. The difference of their genuses is 2.We organize this paper as follow. In Section 2, we illustrate the constructionsof pseudo-vertical surfaces and pseudo-horizontal surfaces. We also study someproperties of a function N (2 k, q ). The value of such a function is the minimalgenus of the closed connected embedded non-orientable surface in the lens space L (2 k, q ) representing the only non-zero Z -homology class. In Section 3, wegive the criterion for the existence for pseudo-horizontal surfaces in small Seifertfibered manifolds. We calculate the genuses and the Z -homology classes of thesesurface. In Section 4, we give several interesting examples of pseudo-horizontalsurfaces. In Section 5, we give an algorithm to calculate the Z -Thurston normfor every non-zero Z -homology class in small Seifert manifolds. SEUDO-HORIZONTAL SURFACES AND Z -THURSTON NORMS 3 Pseudo-vertical and pseudo-horizontal surfaces
Notations for small Seifert fibered manifolds.
We only consider closedorientable Seifert fibered manifolds whose orbits space is orientable. When aSeifert fibered manifold is small, i.e., it does not contain embedded orientableincompressible surfaces, then: (1) its orbit space is S , (2) it has at most threesingular fibers, (3) its Z -homology group is finite. The cases with at most twosingular fibers are lens spaces and well-understood. For the cases with threesingular fibers, we fix the following notations and recall some of the known results.For details, see [8, Chapter 10] or [3, Chapter 2]. • B : a genus 0 compact orientable surface with 3 boundaries. • B : the result of capping off all the boundaries of B by three disks. • M : B × S . • T i ( i = 1 , , i -th boundary of ∂M , homeomorphic to a torus. • v i ( i = 1 , , T i that goes along the S direc-tion. The letter “v” stands for “vertical”. • h i ( i = 1 , , T i that lies on a fixed horizontalsection B × { x } and satisfying [ h ] + [ h ] + [ h ] = 0 ∈ H ( M ; Z ). Theletter “h” stands for “horizontal”. • R i ( i = 1 , , R i is a solid torus. • m i ( i = 1 , , ∂R i that bounds a disk in R i .The letter “m” stands for “meridian”. • l i ( i = 1 , , ∂R i that intersects with m i once.The letter “l” stands for “longitude”. • f i ( i = 1 , , ∂R i → ∂M . • α i , β i , γ i , δ i ( i = 1 , , f i : [ m i ][ l i ] ! α i β i γ i δ i ! [ h i ][ v i ] ! , f − i : [ h i ][ v i ] ! δ i − β i − γ i α i ! [ m i ][ l i ] ! where α i δ i − β i γ i = 1 and α i = 0. • M : M = M S f R S f R S f R . M can be fibered along the S direction. The fibration can extend into thesolid tori R i ’s. Inside R i there is a singular fiber with multiplicity α i . Thegluing result M has a Seifert fibered structure. Following the notation in Martelli[8], we denote it as S (( α , β ) , ( α , β ) , ( α , β )). When M is small, we have P β i /α i = 0, otherwise there will be an incompressible horizontal surface in M . XIAOMING DU
In the literatures, such an S (( α , β ) , ( α , β ) , ( α , β )) has different notations.In Hatcher [3], its notation is M (+0 , β /α , β /α , β /α ). In Orlik [10], itsnotation is [ e, ( o , α , β ′ ) , ( α , β ′ ) , ( α , β ′ )], where (1) e is an integer satisfying e + P β ′ i /α i = P β i /α i and (2) for each i we have α i > < β ′ i < α i . Insome literatures it is also denoted as ( O, o, | e : ( α , β ′ ) , ( α , β ′ ) , ( α , β ′ )), where e, β ′ i are the same as above.2.2. Geometric incompressible surfaces in a solid torus.
Suppose R isa solid torus D × S , m is the meridian circle on ∂R , l is the circle on ∂R that intersect m once. Bredon and Wood [1, Section 6,7,8] and Rubinstein [11,Theorem 13] treated the geometric incompressible surfaces (which can be one-sided) in R . Their results are the followings. Proposition 2.1 (Bredon-Wood, Rubinstein) . Let
R, m, l be as previous. (1) If S is a geometric incompressible surface in R , ∂S lies on ∂R , and [ ∂S ] = p [ l ] + q [ m ] ( p is coprime to q ), then p must be even. (2) On ∂R , supposce c is a simple closed curve whose isotopy class is k [ l ] + q [ m ] ( k = 0 , q is coprime to k ). Then inside R , up to isotopy, there ex-ists a unique geometric incompressible one-sided surface S with boundarycurve as c . Proposition 2.2 (Bredon-Wood) . Under the condition of the previous propo-sition. Denote the genus of the geometric incompressible one-sided surface by N (2 k, q ) ( k > , k > q > . Then N (2 k, q ) is given by the following two equiva-lent ways: (1) Recursive formula: N (2 k,
1) = k , N (2 k, q ) = N (2( k − Q ) , q − m ) + 1 ,where Q, m satisfy km − Qq = ± and < Q < k . (2) Suppose k/q can be written as a continued fraction k/q = [ a , a , . . . , a n ]= a + 1 a + 1 a + 1 . . . + 1 a n where a i ’s are integers, a ≥ , a i > for ≤ i ≤ n , and a n > . Add a i successively except that when a a partial sum is even we skip the next a i . SEUDO-HORIZONTAL SURFACES AND Z -THURSTON NORMS 5 That is, define inductively b = a , b i = ( a i , b i − = a i − or P i − j =0 b j is odd , , b i − = a i − and P i − j =0 b j is even . Then N (2 k, q ) = 12 n X i =0 b i . Remark 2.3.
By the genus of a non-orientable surface with boundaries, we meanthe genus of the closed non-orientable surface obtained by capping off each com-ponent of the boundary with a disk.
Remark 2.4.
In the definition of N (2 k, q ) , the condition k = 0 can be removed ifwe define N (0 ,
1) = 0 since the meridian m bounds a disk in R . In fact, N (2 k, q ) is also the minimal genus of the closed connected embedded non-orientable surfacein the lens space L (2 k, q ) representing the only non-zero Z -homology class. Remark 2.5.
By the homeomorphism classification for lens space, when k, q doesnot satisfy k > or k > q > , we can use the followings to normalize them: (1) N (2 k, q ) = N ( − k, − q ) (hence we can normalize such that k > ). (2) N (2 k, q ) = N (2 k, q + 2 k ) (hence we can normalize such that k > q > ). (3) N (2 k, q ) = N (2 k, k − q ) (hence we can normalize such that k > q > ).We also have N (2 k, q ) = N (2 k, r ) , where qr ≡ k ) . Remark 2.6.
For h > , k > q > and (2 k, q ) = 1 , by the continue fractionexpansion in Proposition 2.2 we easily have (1) N (2 k, q ) ≥ j kq k ≥ . (2) N (2 hq + 2 k, q ) = h + N (2 k, q ) . Remark 2.7.
We can also use the recursive formulae in Remark 2.5 and 2.6 tocalculate N (2 k, q ) in a more explicit way. For example, N (46 ,
7) = N (42+4 ,
7) =3 + N (4 ,
7) = 3 + N (4 ,
3) = 3 + N (4 ,
1) = 5 . Lemma 2.8.
Suppose: (1) c , c are two parallel simple closed curves on ∂R whose isotopy class is p [ l ] + q [ m ] , (2) p = 0 and p is even, (3) S and S are twoone-sided surface inside R with ∂S = c , ∂S = c . Then S T S = ∅ .Proof. The fundamental group of R is Z . Take the covering space e R correspond-ing to the subgroup 2 Z . Denote the lifting of S , S , m, l in e R by e S , e S , e m, e l respectively. Then e S is a two-sided surface and ∂ e S is two simple closed curve XIAOMING DU in the homology class p [ e l ] + q [ e m ]. e R \ e S is disconnected. Both sides has pointsin ∂ e S . Hence e S T e S = ∅ . (cid:3) Lemma 2.9.
Suppose k, q, u, v are fixed integers, where k > q > . When theabsolute value of an integer s is sufficiently large, we have N (2 k · s + u, q · s + v ) ≥ N (2 k, q ) .Proof. Write down the continued fractions of (2 k · s + u ) / ( q · s + v ) and 2 k/q .Suppose 2 k/q = [ a , a , . . . , a n ] . When s → + ∞ , by Euclidean algorithm we know the first n + 1 terms of (2 k · s + u ) / ( q · s + v ) is the same as 2 k/q , i.e.,(2 k · s + u ) / ( q · s + v ) = [ a , a , . . . , a n , . . . ] . By Proposition 2.2 we get the result. When s → −∞ , since N (2 k · s + u, q · s + v ) = N (2 k · ( − s ) − u, q · ( − s ) − v ), the result still holds. (cid:3) Pseudo-vertical and pseudo-horizontal surface.
Suppose M is a Seifertfibered manifold. A surface in M is called vertical if it is consist of regular fibers.A surface in M is called horizontal if it intersect each regular fiber. A connectedsurface S in M is called pseudo-vertical if: (1) S ∩ M is a vertical annulus whoseboundary lies in two distinct T i and T j , and (2) S ∩ R i and S ∩ R j are one-sidedgeometric incompressible surfaces inside the solid torus R i and R j respectively.For simplicity, we also called it the pseudo-vertical surface connecting R i and R j .We denote such a pseudo-vertical surface as V i,j . A one-sided surface S in M iscalled pseudo-horizontal if S ∩ M is horizontal in M and S ∩ R i is either afamily of meridian disks or a one-sided geometric incompressible surface in R i foreach i . Remark 2.10.
The pseudo-horizontal surfaces have a more geometric descrip-tion. Jaco [4] constructed what is now called “staircase” surfaces in M = B × S .Then the pseudo-horizontal surfaces are obtained by attaching either disks or asingle non-orientable surface in a solid torus to such a staircase surface. A moreconcrete picture in a specific small Seifert fibered manifold is shown in Figure 2after Example 4.1 of our paper. Frohman [2, Theorem 2.5] proved a structure theorem for the geometric incom-pressible one-sided surfaces in Seifert manifolds.
SEUDO-HORIZONTAL SURFACES AND Z -THURSTON NORMS 7 Theorem 2.11 (Frohman) . Every closed one-sided geometric incompressible sur-face in a Seifert fibered manifold M is isotopic to a pseudo-vertical surface or apseudo-horizontal surface. Remark 2.12.
The converse of the above theorem is not true. There are pseudo-vertical surfaces and pseudo-horizontal surfaces which are compressible. Frohmanin his paper gave the first example that in S ((2 , − , (3 , , (6 , the pseudo-vertical surface is compressible. This Seifert fibered manifold is not small. Someof the examples in small Seifert fibered manifolds are in Section 4 of our paper. The homology classes of the pseudo-vertical surfaces and pseudo-horizontal surfaces.
We follow the notation at the beginning of Section 2. All v i ’s are in the same homology class. Hence we denote their homology class by [ v ].We only consider the Z -homology groups. The following results can be verifiedby direct computation or geometric construction. Lemma 2.13.
The Z -homology group H ( M ; Z ) is H ( M ; Z ) = h [ h ] , [ h ] , [ h ] , [ v ] | α [ h ] + β [ v ] = 0 ,α [ h ] + β [ v ] = 0 ,α [ h ] + β [ v ] = 0 , [ h ] + [ h ] + [ h ] = 0 i . Lemma 2.14.
About H ( M ; Z ) and the representing circle. (1) If α , α , α are odd, then (1.1) If β + β + β is odd, then H ( M ; Z ) = 0 . (1.2) If β + β + β is even, then we can use fiber moves to make all β i ’seven so that H ( M ; Z ) = Z , [ v ] = 0 , [ h ] = [ h ] = [ h ] = 0 . (2) If there is only one even α i in { α , α , α } , then H ( M ; Z ) = 0 . (3) If there is exactly two even α i in { α , α , α } , then H ( M ; Z ) = Z .Suppose α and α are even. Then [ h ] = [ h ] = 0 , [ v ] = 0 , and whether [ h ] equals depends on β is even or odd. (4) If α , α , α are even, then H ( M ; Z ) = Z ⊕ Z , and [ h ] , [ h ] , [ h ] rep-resent the three non-zero elements in H ( M ; Z ) , [ v ] = 0 . By duality, H ( M ; Z ) ∼ = H ( M ; Z ). Lemma 2.15.
Each of the non-zero elements in H ( M ; Z ) can be representedby a pseudo-horizontal surface or a pseudo-vertical surface. More specifically, XIAOMING DU (1) If α , α , α are odd and β , β , β are even, then there is no pseudo-verticalsurface and the only non-zero element in H ( M ; Z ) can be represented bya pseudo-horizontal surface. (2) If exactly two of α , α , α are even, then the only non-zero element in H ( M ; Z ) can be represented by a pseudo-vertical surface that connectsthe two R i ’s corresponding to the two even α i ’s. (3) If α , α , α are even, suppose { i, j, k } = { , , } and denoted the pseudo-vertical surface connecting R i and R j by V i,j , then V , , V , , V , representdifferent non-zero elements in H ( M ; Z ) and for each i, j ( i = j ) we have [ V i,j ] · [ h k ] = 0 , [ V i,j ] · [ h i ] = 1 , [ V i,j ] · [ h j ] = 1 . The conditions for the existence of pseudo-horizontal surfacesin small Seifert manifolds
We fix the notation for pseudo-horizontal surfaces in small Seifert manifolds asfollow. • Z : a pseudo-horizontal surface in M . • Z : Z T M . There is a covering map from Z to B . • c i ( i = 1 , , ∂Z T T i . • λ i , µ i ( i = 1 , , λ i > c i ] = (cid:16) λ i µ i (cid:17) [ h i ][ v i ] ! .Since ( λ i , µ i ) and ( − λ i , − µ i ) correspond to the same curve, we can alwaysassume λ i >
0. But µ i is allowed to be <
0. We also call Z “the pseudo-horizontal surface determined by ( λ , µ ) , ( λ , µ ) , ( λ , µ )”. • λ : the least common multiple of λ , λ , λ . It equals to the covering degreeof Z → B . The number of the components of ∂Z ∩ T i is λ/λ i . Proposition 3.1. µ /λ + µ /λ + µ /λ = 0 .Proof. The same as the case of horizontal surfaces, cf [13] or [3, Proposition1.11]. (cid:3)
Proposition 3.2. λ i ’s and µ i ’s satisfy one of the followings. (1) All λ i ’s are odd and µ + µ + µ is even. (2) Exactly two of λ i ’s are even. (3) All λ i ’s are even and all µ i ’s are odd.Proof. By direct computation in elementary number theory. (cid:3)
SEUDO-HORIZONTAL SURFACES AND Z -THURSTON NORMS 9 Proposition 3.3.
For each i , one of the following two cases happens. (1) ( λ i , µ i ) = ( α i , β i ) . (2) λ i equals λ (recall λ is the least common multiple of λ , λ , λ ).Proof. If λ/λ i >
1, then Z T R i must be λ/λ i copies of geometric incompressiblesurfaces inside R i and bounded by λ/λ i simple closed curves that parallel to c i .If c i is not the meridian, by Lemma 2.8, these surfaces must intersect and Z isnot embedded. (cid:3) Proposition 3.4.
If there exists a pseudo-horizontal surface determined by ( λ , µ ) , ( λ , µ ) , ( λ , µ ) in a Seifert manifold M = S (( α , β ) , ( α , β ) , ( α , β )) , thenfor each i , we have | ( λ i − α i ) and | ( µ i − β i ) .Proof. If λ/λ i >
1, then ( λ i , µ i ) = ( α i , β i ), the results of the proposition holds.If λ/λ i = 1, then the homology class of c i on ∂R i is ( λ i δ i − µ i γ i )[ m i ] + ( − λ i β i + µ i α i )[ l i ]. It can bound a geometric incompressible surface if and only if − λ i β i + µ i α i is even. Since there is at most one of α i , β i is even, and at most one of λ i , µ i is even, we have one of the following cases happen:(1) λ i odd, β i even, µ i even, α i odd.(2) λ i even, β i odd, µ i odd, α i even.(3) λ i odd, β i odd, µ i odd, α i odd.In each case, the result of the proposition holds. (cid:3) Theorem 3.5.
There exists a pseudo-horizontal surface determined by ( λ , µ ) , ( λ , µ ) , ( λ , µ ) in a Seifert manifold M = S (( α , β ) , ( α , β ) , ( α , β )) if andonly if the results from Proposition 3.1 to Proposition 3.4 hold.Proof. The “only if” part is the above propositions. For the “if” part, since( α i , β i ), ( λ i , µ i ) satisfy these conditions, we can first construct a horizontal surfacein M and then extend it into each R i to construct the pseudo-horizontal surface. (cid:3) Proposition 3.6.
The Z -homology class of a pseudo-horizontal surface Z whichis determined by ( λ i , µ i ) ’s is described as follows. (1) When H ( M ; Z ) = Z , Z represents the only non-zero Z -homology class. (2) When H ( M ; Z ) = Z ⊕ Z , the intersection number of [ Z ] and [ h i ] is µ i · λ/λ i . Hence we can detect the Z -homology class that Z represents by comparing its intersection numbers with [ h ] , [ h ] , [ h ] to those of thepseudo-vertical surfaces V , , V , , V , .Proof. Direct geometric construction. (cid:3)
Proposition 3.7.
The genus of the pseudo-horizontal surface Z determined by ( λ i , µ i ) ’s is λ (cid:20) − ( 1 λ + 1 λ + 1 λ ) (cid:21) + X i =1 N ( − λ i β i + µ i α i , λ i δ i − µ i γ i ) . Proof.
Suppose Z = Z ∩ M . If we cap off all the boundaries of Z with disks,then the Euler characteristic of the resulting closed surface can be compute byRiemann-Hurwitz formula as λ " − X i =1 (1 − λ i ) = λ (cid:20) ( 1 λ + 1 λ + 1 λ ) − (cid:21) . Its genus is two minus the Euler characteristic. Then we replace the disks by theone-sided surfaces in R i ’s and then get the result. (cid:3) Examples.
In this section, we first give three families of small Seifert fibered manifolds inwhich there exist pseudo-vertical surfaces that is not Z -taut. Then in the nextexample, the pseudo-vertical surface is isotopic to a pseudo-horizontal surface.In the final example, all the pseudo-horizontal surfaces are not Z -taut and thethree pseudo-vertical surfaces are Z -taut. Example 4.1. M = S ((2 , − , (2 m + 1 , m ) , (2 n, , where n > m + 1 . In this example, α β γ δ ! = − − ! , α β γ δ ! = m + 1 m ! , α β γ δ ! = n n − ! . Let ( λ , µ ) = (2 , − λ , µ ) = (2 m + 1 , m ), and ( λ , µ ) = (4 m + 2 , λ = 4 m + 2, the genus of the pseudo-horizontal surface is2 + (4 m + 2) (cid:18) − − m + 1 − m + 2 (cid:19) + N ( − (4 m + 2) + 2 n, (4 m + 2) − (2 n − m + N (2 n − (4 m + 2) ,
1) = 2 m + n − (2 m + 1) = n − . The genus of the pseudo-vertical surface in the same Z -homology class is n + 1. SEUDO-HORIZONTAL SURFACES AND Z -THURSTON NORMS 11 Figure 1 shows the attaching way from each ∂R i to ∂M for the small Seifertmanifold M = S ((2 , − , (3 , , (8 , M minus a tubular neighbourhoodof a regular fiber, denoted as M ′ . The right Figure 1 is three solid tori. We drawthe corresponding curves for the attaching. M v h h h m m m l l l R R R Figure 1.Figure 2 shows a pseudo-horizontal surface determined by ( λ , µ ) = (2 , − λ , µ ) = (3 , λ , µ ) = (6 ,
1) inside S ((2 , − , (3 , , (8 , M ′ . The middle of Figure 2 is the back part of M ′ . The right of Figure 3 shows five disks and one M¨obius band in the solidtori R , R , R . We can see the staircase surface in M ′ . The genus of such apseudo-horizontal surface is 3, while the genus of the pseudo-vertical surface V , connecting R and R is 5. Figure 2. Example 4.2. M = S ((3 , − , (4 , , (2 n, , where n > . In this example, α β γ δ ! = − − ! , α β γ δ ! = ! , α β γ δ ! = n n − ! . Let ( λ , µ ) = (3 , − λ , µ ) = (4 , λ , µ ) = (12 , λ = 12, thegenus of the pseudo-horizontal surface is2 + 12 (cid:18) − − − (cid:19) + N ( −
12 + 2 n, − n + 1) = 6 + N (2 n − ,
1) = n. The genus of the pseudo-vertical surface V , which is in the same Z -homologyclass is n + 2. Example 4.3. M = S (( m, − , (2 n , , (2 n , where m ≥ , n ≥ m , n ≥ m , and n + n > m . In this example, α β γ δ ! = m − − m ! , α β γ δ ! = n n − ! , α β γ δ ! = n n − ! . Let ( λ , µ ) = ( m, − λ , µ ) = (2 m, λ , µ ) = (2 m, λ = 2 m , thegenus of the pseudo-horizontal surface is2 + 2 m (cid:18) − m − m − m (cid:19) + N (2 n − m,
1) + N (2 n − m,
1) = n + n − . Such a pseudo-horizontal surface intersects h and h once respectively. Hence thepseudo-vertical surface V , that connects R and R is in the same Z -homologyclass as this pseudo-horizontal surface. The genus of this pseudo-vertical surfaceis n + n . Remark 4.4.
There is another method to estimate the lower bound for the Z -Thurston norm for the non-zero elements in H ( M ; Z ) . It uses Heegaard Floerhomology and calculates the correction term. Using the algorithm in [9, 14] andan online program given by [7] , we try to compute the lower bound of the Z -Thurston norm for the previous examples when the parameters m, n, n , n arenot too large. We find the results in our examples meet the lower bounds of the Z -Thurston norms. Example 4.5. M = S ((2 , − , (3 , , (4 , . In this example, α β γ δ ! = − − ! , α β γ δ ! = ! , α β γ δ ! = ! . SEUDO-HORIZONTAL SURFACES AND Z -THURSTON NORMS 13 Let ( λ , µ ) = (2 , − λ , µ ) = (3 , λ , µ ) = (6 , λ = 6, the genusof the pseudo-horizontal surface is2 + 6 (cid:18) − − − (cid:19) + N ( − · · , · − ·
3) = 2 + N (2 ,
1) = 3 . The genus of the pseudo-vertical surface V , connecting R and R is also 3. By[12], the mapping class group of such an M is trivial. Hence the pseudo-horizontalsurface is isotopic to the pseudo-vertical surface. Example 4.6. M = S ((2 , − , (2 , , (2 n, , where n > . In this example, α β γ δ ! = − − ! , α β γ δ ! = ! , α β γ δ ! = n n − ! . As before, denote the pseudo-vertical surface connecting R i and R j by V i,j . V , is a Klein bottle and hence V , must be Z -taut. In fact, if there exists anembedded R P , then its regular neighbourhood will give a connected sum piecehomeomorphic to R P , which contradict to the structure of M . Both V , and V , have genus n + 1. We will prove they are also Z -taut.At first we find the possibilities of ( λ i , µ i ). By Proposition 3.4, all λ i ’s are even.Suppose { i, j, k } = { , , } and λ i ≤ λ j ≤ λ k . According to their relations wehave four cases.(1) If λ i = λ j = λ k , since all λ , λ , λ are even, all µ , µ , µ are odd. Thiscontradicts to µ /λ + µ /λ + µ /λ = 0.(2) If λ i = λ j < λ k , since µ /λ + µ /λ + µ /λ = 0, this would not happen.(3) If λ i < λ j = λ k , then λ i is not the least common multiple of λ i , λ j , λ k . ByProposition 3.3, ( λ i , µ i ) = ( α i , β i ). This case might happen.(4) If λ i < λ j < λ k , then similarly ( λ i , µ i ) = ( α i , β i ) and ( λ j , µ j ) = ( α j , β j ).Now λ i < λ j implies ( λ i , µ i ) = (2 , −
1) or (2 , µ /λ + µ /λ + µ /λ = 0 we get λ j = λ k , which contradict to λ i < λ j < λ k .The only possible case (3) has three subcases:(3.1) i = 1, ( λ i , µ i ) = ( α , β ) = (2 , − i = 2, ( λ i , µ i ) = ( α , β ) = (2 , i = 3, ( λ i , µ i ) = ( α , β ) = (2 n, Z -homology class of the pseudo-horizontal surface deter-mined by ( λ , µ ) , ( λ , µ ) , ( λ , µ ) in each subcase. Subcase (3.1): By Proposition 3.1, Proposition 3.4 and λ < λ = λ we canassume ( λ , µ ) = (2 p, s ) and ( λ , µ ) = (2 p, p − s ), where p > , ( s, p ) = 1, here s can be less than zero. The least common multiple λ of λ , λ , λ is 2 p . p − s is coprime to 2 p implies p must be even. By Proposition 3.6, the intersectionnumbers of such a pseudo-horizontal surface Z with the 1-dimensional homologyclasses are: [ Z ] · [ h ] = p , [ Z ] · [ h ] = s , [ Z ] · [ h ] = p − s . Hence Z is in the same Z -homology class as V , .Subcase (3.2): It is similar to subcase (3.1). Z is in the same Z -homologyclass as V , .Subcase (3.3): By Proposition 3.1, Proposition 3.4 and λ < λ = λ , wecan assume ( λ , µ ) = (2 p, s ) and ( λ , µ ) = (2 p, t ), where p > n , ( s, p ) = 1,( t, p ) = 1, and n + s p + t p = 0. Now s + t = − p/n , s and t are odd, so p/n and p are even. By Proposition 3.6, the intersection number of such a pseudo-horizontalsurface Z with the 1-dimensional homology class is: [ Z ] · [ h ] = s , [ Z ] · [ h ] = t ,[ Z ] · [ h ] = p/n . Hence Z is in the same Z -homology class as V , . Since V , isalready Z -taut, the pseudo-horizontal surface cannot have a smaller genus.Finally we calculate the genus of the pseudo-horizontal surface in subcase (3.1).By Proposition 3.7, the genus of the pseudo-horizontal surface is2 + 2 p (cid:18) − − p − p (cid:19) + N ( − p + s · , p − s )+ N ( − p + ( p − s ) · n, p − ( p − s ) · (2 n − p + N (2 p − s, s ) + N (2 n · ( p − s ) − p, p − s ) . When p ≥ n , the genus of the pseudo-horizontal surface is larger than n + 1and hence is not taut. It only remain for p < n .(i) If p − s <
0, then by Corollary 2.6 we have N (2 n · ( p − s ) − p, p − s ) = N (2 n · ( s − p ) + 2 p, s − p ) = n + N (2 p, s − p ) ≥ n + 1.(ii) If p − s >
0, substitute p − s by t , then by Corollary 2.6 we have N (2 n · ( p − s ) − p, p − s ) = N (2 nt − p, t ) = N ((2 n − p ) · t + 2 p · ( t − , t ) =( n − p ) + N (2 p ( t − , t ) ≥ n − p + 1.In each case the genus of the pseudo-horizontal surface is larger than n +1. So thepseudo-horizontal surfaces are always not Z -taut and hence the pseudo-verticalsurfaces are taut. SEUDO-HORIZONTAL SURFACES AND Z -THURSTON NORMS 15 Z -Thurston norms in small Seifert manifolds. Our strategy is as follow: First we prove that when some of the λ i ’s or µ i ’sare sufficiently large, the pseudo-horizontal surface determined by ( λ i , µ i )’s isnot Z -taut. Hence we only need to analyze finitely many pseudo-horizontalsurfaces determined by relatively small λ i ’s or µ i ’s, find their Z -homology classand compare the genuses.We follow the notation at the beginning of Section 2 and the beginning ofSection 3. Without loss of generality, we suppose α ≤ α ≤ α , { i, j, k } = { , , } (as unordered sets), λ i ≤ λ j ≤ λ k . Remark 5.1. µ /λ + µ /λ + µ /λ = 0 does not imply the largest λ k is theleast common multiple of λ i , λ j , λ k . For example + + − = 0 . However,when each of λ i , λ j , λ k is not the least common multiple, by Proposition 3.3, foreach i we have ( λ i , µ i ) = ( α i , β i ) . Then we can only cap off the staircase surfaceby disks. The result is an orientable horizontal surface, not a non-orientablepseudo-horizontal surface. This cannot happen in small Seifert manifolds. Hencefor pseudo-horizontal surfaces we can always assume the largest λ k to be the leastcommon multiple of λ i , λ j , λ k . Similar to Example 4.6, according to the relations between λ i , λ j , λ k , we havefour cases: (1) λ i = λ j = λ k ; (2) λ i = λ j < λ k ; (3) λ i < λ j = λ k ; (4) λ i < λ j < λ k .We will analyze them one by one. Case (1): λ i = λ j = λ k = λ . In this case, we have µ + µ + µ = 0.Hence λ must be odd. By Proposition 3.4, it only happens when all α i ’s are odd.By Proposition 2.15 there is no pseudo-vertical surface. Each pseudo-horizontalsurface intersects at λ points with the regular fiber. So each pseudo-horizontalsurface represents the only non-zero element in H ( M ; Z ). By Proposition 3.7,the genus of the pseudo-horizontal surface is λ − X i =1 N ( − λβ i + µ i α i , λδ i − µ i γ i ) . For a given small Seifert manifold S (( α , β ) , ( α , β ) , ( α , β )), since λ, µ i , µ j , µ k can vary, the genus of the pseudo-vertical surface can vary. Now let us useProposition 3.7 to see when can we minimize the genus.Fist let λ = 1. From the given µ , µ , µ satisfying µ + µ + µ = 0 we constructa pseudo-horizontal surface. When µ i is sufficiently large, by Lemma 2.9 we have N ( − β i + µ i α i , δ i − µ i γ i ) ≥ N ( α i , γ i ). By Remark 2.5 and α i δ i − β i γ i = 1, we have N ( α i , γ i ) = N ( α i , β i ). Hence the genus of the pseudo-horizontal surface is at least N ( α , β ) + N ( α , β ) + N ( α , β ). So we can check finitely many µ i , µ j , µ k thatis not to large, compare the genuses, and then determine the minimal one when λ = 1.When λ is larger than the above minimal genus when λ = 1, the genus of thepseudo-horizontal surface becomes larger. So we only need to try finitely many λ . For each fixed λ , similarly, when µ i is sufficiently large, N ( − λβ i + µ i α i , λδ i − µ i γ i ) ≥ N ( α i , γ i ) = N ( α i , β i ). Hence again we can only try finitely many µ i , µ j , µ k which is not too large to determine the minimal genus. Case (2): λ i = λ j < λ k . By µ /λ + µ /λ + µ /λ = 0, this case cannothappen. Case (3): λ i < λ j = λ k . In this case, λ j and λ k must be the least commonmultiple λ and ( λ i , µ i ) = ( α i , β i ). Suppose λ j = λ k = pλ i . Since µ /λ + µ /λ + µ /λ = 0, pµ i + µ j + µ k = 0. This case can happen. According to whetherthese parameters are odd or even, we have three subcases and can check the Z -homology classes of the pseudo-vertical surfaces and the pseudo-horizontalsurfaces in each subcase.(3.1) λ i is even, λ j = λ k = pλ i is even. Then µ i , µ j , µ k are odd. By pµ i + µ j + µ k = 0, p is even. By Proposition 3.4, α i , α j , α k are even. So β i , β j , β k areodd. The pseudo-horizontal surface intersects odd times with h j , h k andhence lies in the same Z -homology class as the pseudo-vertical surface V j,k .(3.2) λ i is odd, λ j = λ k = pλ i is even. Then p is even, µ j , µ k are odd. ByProposition 3.4, α i is odd, α j , α k are even. The pseudo-horizontal surfaceintersects odd times with h j , h k and hence lies in the same Z -homologyclass as the pseudo-vertical surface V j,k .(3.3) λ i is odd, λ j = λ k = pλ i is odd. Then p is odd. By Proposition 3.4, α i , α j , α k are odd. There is no pseudo-vertical surface. H ( M ; Z ) = Z . Each pseudo-horizontal surface lies in the only non-zero Z -homologyclass.For a given small Seifert manifold S (( α , β ) , ( α , β ) , ( α , β )), now ( λ i , µ i ) hasonly three choices: ( α , β ), ( α , β ), and ( α , β ). But λ j , µ j , λ k , µ k can vary, thegenus of the pseudo-vertical surface can vary. Now let us use Proposition 3.7 tosee when can we minimize the genus. SEUDO-HORIZONTAL SURFACES AND Z -THURSTON NORMS 17 Subcase (3.1): The genus of the pseudo-vertical surface V j,k is N ( α j , β j ) + N ( α k , β k ). The genus of the pseudo-horizontal surface is λ − λλ i + N ( − λβ j + µ j α j , λδ j − µ j γ j ) + N ( − λβ k + µ k α k , λδ k − µ k γ k ) . When λ − λ/λ i ≥ N ( α j , β j ) + N ( α k , β k ), the pseudo-horizontal surface does nothave the minimal genus. We only need to try a finite number of small λ to detectwhether pseudo-horizontal surfaces have the minimal genus.For a fixed λ , when µ j → ±∞ , again, we have N ( − λβ j + µ j α j , λδ j − µ j γ j ) ≥ N ( α j , γ j ) = N ( α j , β j ) and similarly N ( − λβ k + µ k α k , λδ k − µ k γ k ) ≥ N ( α k , β k ).Hence these pseudo-horizontal surfaces do not have minimal genus. So we cancheck finitely many µ j , µ k that is not to large and compare the genuses.Subcase (3.2): Similar to subcase (3.1).Subcase (3.3): The genus of the pseudo-horizontal surface is λ − λλ i + N ( − λβ j + µ j α j , λδ j − µ j γ j ) + N ( − λβ k + µ k α k , λδ k − µ k γ k ) . Similar to Case (1).
Case (4): λ i < λ j < λ k . Now λ i , λ j is not the least common multiple and λ k is the least common multiple. So ( λ i , µ i ) = ( α i , β i ), ( λ j , µ j ) = ( α j , β j ), i < j ,and α i < α j . The rest ( λ k , µ k ) can be determined by µ i /λ i + µ j /λ j + µ k /λ k = 0.So the genus of the pseudo-horizontal surface can be completely determined by( α i , β i ) , ( α j , β j ) , ( α k , β k ). Algorithm.
Finally, given the small Seifert manifold S (( α , β ) , ( α , β ) , ( α , β ))(assuming α ≤ α ≤ α ), we summarize the algorithm for the Z -Thurston normsas follow.Step 1. If α < α , take ( λ , µ ) = ( α , β ) and ( λ , µ ) = ( α , β ). Calculate( λ , µ ) from µ /λ + µ /λ + µ /λ = 0. If λ < λ , take i = 1 , j = 2 , k = 3, tryCase (4). If α < α , take ( λ , µ ) = ( α , β ) and ( λ , µ ) = ( α , β ). Calculate( λ , µ ) from µ /λ + µ /λ + µ /λ = 0. If λ < λ , take i = 1 , j = 3 , k = 2, tryCase (4). If α < α , take ( λ , µ ) = ( α , β ) and ( λ , µ ) = ( α , β ). Calculate( λ , µ ) from µ /λ + µ /λ + µ /λ = 0. If λ < λ , take i = 2 , j = 3 , k = 1, tryCase (4).Step 2. Take i = 1, ( λ , µ ) = ( α , µ ), let λ < λ = λ , try Case (3).Take i = 2, ( λ , µ ) = ( α , µ ), let λ < λ = λ , try Case (3). Take i = 3,( λ , µ ) = ( α , µ ), let λ < λ = λ , try Case (3). Step 3. If α , α , α are odd, try Case (1).In each step, we only need to try finitely many times. So we can determine theminimal genus. References [1] G. E. Bredon, J. W. Wood. Non-orientable surfaces in orientable 3-manifolds. Invent. Math.7(1969) 83–110.[2] C. Frohman. One-sided incompressible surfaces in Seifert fibered spaces. Topology Appl.23 (1986), no. 2, 103–116.[3] A. Hatcher. Notes on basic 3-manifold topology (online notes, unpublished).http://pi.math.cornell.edu/˜hatcher/3M/3Mdownloads.html[4] W. Jaco. Surfaces embedded in M × S . Can. J. Math., Vol. XXII, No. 3, 1970, pp.553-568.[5] W. Jaco, J. H. Rubinstein, S. Tillmann. Z -Thurston norm and complexity of 3-manifolds.Math. Ann. 356 (2013), no. 1, 1–22.[6] W. Jaco, J. H. Rubinstein, J. Spreer, S. Tillmann. Z -Thurston norm and complexity of3-manifolds, II. Algebraic and Geometric Topology, 20 (2020), no. 1, 503–529.[7] C. Karakurt’s webpage. http://web0.boun.edu.tr/cagri.karakurt/Research.html[8] B. Martelli. An Introduction to Geometric Topology. CreateSpace Independent PublishingPlatform, 2016.[9] Y. Ni, Z. Wu. Correction terms, Z -Thurston norm, and triangulations. Topology Appl.194(2015), 409–426.[10] P. Orlik. Seifert Manifolds. Lecture Notes in Mathematics 291, Springer-Verlag 1972.[11] J. H. Rubinstein. One-sided Heegaard splitting of 3-manifolds. Pacific Journal of Mathe-matics, Vol. 76, No. 1, 1978.[12] J. H. Rubinstein and J. Birman. One-sided Heegaard splittings and homeotopy groups ofsome 3-manifolds. Proc. London Math. Soc. (3) 49 (1984), no. 3, 517–536.[13] F. Waldhausen. Eine Klasse von 3-dimensionalen Mannigfaltigkeiten I, II, Invent. Math.3, 308–333 (1967)[14] Z. Wu and J. Yang. Studies of distance one surgeries on lens space L ( p, School of Mathematics, South China University of Technology, Guangzhou,510640, P. R. China
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