
Let M be a compact connected orientable 3-manifold and F be a compact connected orientable surface properly embedded in M. If F cuts M into two handlebodies X and Y (i.e., M=X∪FY), then we say that F is an H′-splitting surface for M and call X∪FY an H′-splitting for M. When the H′-splitting surface F is incompressible in a handlebody H, a characteristic of an H′-splitting H1∪FH2 to denote H is already known. In the present paper, we generalize the above result as follows: Let H be a handlebody of genus g≥1, X∪FY an H′-splitting for H. Then, either X∪FY is stabilized, or there exists a reducing system J1∪K1 of F, such that J1 is quasi-primitive in Y and K1 is quasi-primitive in X. Combining the result with the known result, we obtain a characteristic of an H′-splitting H1∪FH2 to denote a handlebody.
Citation: Yan Xu, Bing Fang, Fengchun Lei. On H′-splittings of a handlebody[J]. AIMS Mathematics, 2024, 9(9): 24385-24393. doi: 10.3934/math.20241187
[1] | Mourad Berraho . On a problem concerning the ring of Nash germs and the Borel mapping. AIMS Mathematics, 2020, 5(2): 923-929. doi: 10.3934/math.2020063 |
[2] | Wenlong Sun, Gang Lu, Yuanfeng Jin, Choonkil Park . Self-adaptive algorithms for the split problem of the quasi-pseudocontractive operators in Hilbert spaces. AIMS Mathematics, 2022, 7(5): 8715-8732. doi: 10.3934/math.2022487 |
[3] | Yajun Xie, Changfeng Ma . The hybird methods of projection-splitting for solving tensor split feasibility problem. AIMS Mathematics, 2023, 8(9): 20597-20611. doi: 10.3934/math.20231050 |
[4] | Ting Huang, Shu-Xin Miao . More on proper nonnegative splittings of rectangular matrices. AIMS Mathematics, 2021, 6(1): 794-805. doi: 10.3934/math.2021048 |
[5] | Wenxiu Guo, Xiaoping Lu, Hua Zheng . A two-step iteration method for solving vertical nonlinear complementarity problems. AIMS Mathematics, 2024, 9(6): 14358-14375. doi: 10.3934/math.2024698 |
[6] | Li-Tao Zhang, Xian-Yu Zuo, Shi-Liang Wu, Tong-Xiang Gu, Yi-Fan Zhang, Yan-Ping Wang . A two-sweep shift-splitting iterative method for complex symmetric linear systems. AIMS Mathematics, 2020, 5(3): 1913-1925. doi: 10.3934/math.2020127 |
[7] | Ümit Tokeşer, Tuğba Mert, Yakup Dündar . Some properties and Vajda theorems of split dual Fibonacci and split dual Lucas octonions. AIMS Mathematics, 2022, 7(5): 8645-8653. doi: 10.3934/math.2022483 |
[8] | Junjiang Lai, Zhencheng Fan . Stability for discrete time waveform relaxation methods based on Euler schemes. AIMS Mathematics, 2023, 8(10): 23713-23733. doi: 10.3934/math.20231206 |
[9] | Kaiqing Huang, Yizhi Chen, Aiping Gan . Structure of split additively orthodox semirings. AIMS Mathematics, 2022, 7(6): 11345-11361. doi: 10.3934/math.2022633 |
[10] | Meiying Wang, Luoyi Shi, Cuijuan Guo . An inertial iterative method for solving split equality problem in Banach spaces. AIMS Mathematics, 2022, 7(10): 17628-17646. doi: 10.3934/math.2022971 |
Let M be a compact connected orientable 3-manifold and F be a compact connected orientable surface properly embedded in M. If F cuts M into two handlebodies X and Y (i.e., M=X∪FY), then we say that F is an H′-splitting surface for M and call X∪FY an H′-splitting for M. When the H′-splitting surface F is incompressible in a handlebody H, a characteristic of an H′-splitting H1∪FH2 to denote H is already known. In the present paper, we generalize the above result as follows: Let H be a handlebody of genus g≥1, X∪FY an H′-splitting for H. Then, either X∪FY is stabilized, or there exists a reducing system J1∪K1 of F, such that J1 is quasi-primitive in Y and K1 is quasi-primitive in X. Combining the result with the known result, we obtain a characteristic of an H′-splitting H1∪FH2 to denote a handlebody.
It is a well known fact that each compact connected orientable 3-manifold M admits a Heegaard splitting V∪FW, where F is an orientable closed surface embedded in the interior of M which cuts M into two compression bodies V and W with ∂+V=F=∂+W. In 1970, Downing [1] proved that each compact connected 3-manifold M with nonempty boundary has a decomposition as H1∪FH2, where H1 and H2 are two handlebodies with the same genus, and F=H1∩H2 is a connected surface properly embedded in M. In 1973, Roeling[2] discussed such handlebody-splittings for 3-manifolds with connected boundaries. Later, Suzuki[3] slightly modified the results of Downing[1] and Roeling[2], and formulated a Haken type theorem for these handlebody-splittings in the way of Casson-Gordon[4].
Let M be a compact connected orientable 3-manifold and F be a compact connected orientable surface properly embedded in M. If F cuts M into two handlebodies X and Y (not necessarily with the same genus), that is, M=X∪FY, then we say that F is an H′-splitting surface for M and call X∪FY an H′-splitting of M. It is clear that if M is closed, then the H′-splitting X∪FY is just a Heegaard splitting for M. If M is with non-empty boundary, then the H′-splitting X∪FY is distinct from a Heegaard splitting of M.
It has been shown in [5] that each compact connected orientable 3-manifold admits an H′-splitting (i.e., H′-splittings, similar to Heegaard splittings, which are common structures of 3-manifolds). Thus, it follows that it is a new way to construct all compact connected orientable 3-manifolds.
The Casson-Gordon Theorem [4] on weakly reducible Heegaard splittings has been generalized to the H′-splitting case in [5]. On the other hand, there exist examples (refer to [5]) to show that Haken's lemma does not hold in the H′-splitting case in general. This implies that the properties of H′-splitting structures for the 3-manifolds with boundaries are quite different from the Heegaard splitting structures and Downing's handlebody-splitting structures as above.
A characteristic of an H′-splitting H1∪FH2 to denote a handlebody has been described in [6], where F is incompressible in both H1 and H2 (see Theorem 2.7 in Section 2 or [6] for the detail). In the present paper, we generalize the above result, regardless that F is compressible or incompressible in H, as follows: Let H be a handlebody of genus g≥1, X∪FY an H′-splitting for H. Then, either X∪FY is stabilized or there exists a reducing system J1∪K1 of F, such that J1 is quasi-primitive in Y and K1 is quasi-primitive in X. (refer to Sections 2 and 3 for the definitions). Combining the result with Theorem 2.7, we obtain a characteristic of an H′-splitting X∪FY to denote a handlebody.
The other parts of the paper is organized as follows. In Section 2, some necessary preliminaries are given. In Section 3, the statements of the main results and their proofs are given. It is worth noting that a refined version (see Lemma 2.9) of the Haken's lemma in disk case plays an essential role in the the proof of Theorem 3.3.
In this section, we will review some notions and fundamental facts about 3-manifolds that will be used in Section 3. All the 3-manifolds considered in the paper are assumed to be compact and orientable. The concepts and terminologies which are not defined in the paper are all standard (refer to, for example, [7,8,9]).
Let M be a 3-manifold. A 2-sphere S embedded in M is essential in M if S does not bound a 3-ball in M; otherwise, S is inessential in M. M is reducible if M contains an essential 2-sphere; otherwise, M is irreducible.
Let M be a compact 3-manifold, and F a 2-sided surface properly embedded in M or F⊂∂M. If there exists a disk D⊂M such that D∩F=∂D and ∂D is essential in F, then we say that F is compressible in M. Such a disk D is called a compressing disk of F. F is incompressible if F is not compressible in M and no component of F is an inessential 2-sphere, parallel to a disk in ∂M, or a disk in ∂M. If ∂M is compressible in M, then M is said to be ∂-reducible.
A handlebody H is a 3-manifold such that there exists a collection D={D1,⋯,Dn} of pairwise disjoint disks properly embedded in H such that the manifold obtained by cutting H open along D is a 3-ball. D is called a complete system of disks for H, and n is called the genus of H.
Let S be an orientable closed surface, and J={J1,⋯,Jk} a collection of pairwise disjoint simple closed curves (s.c.c.) on S. Let C be the 3-manifold obtained by adding 2-handles to S×I along J×0, then capping of any resulting 2-spheres with 3-balls. C is called a compression body. In C, set ∂+C=S×1 and ∂−C=∂C−∂+C. J is naturally extended to a collection D={D1,⋯,Dk} of pairwise disjoint disks properly embedded in C. D is called a defining system of disks for C. It is clear that if ∂−C=∅, C is a handlebody; and if ∂−C≠∅, the manifold obtained by cutting C open along D is homeomorphic to ∂−C×I. S×I is called a trivial compression body.
Let D={D1,...,Dk} be a defining disk system for a compression body C, and Δ a disk in C such that for some i, 1≤i≤k, Δ∩Dj=∅ for j≠i, and Δ∩Di=α is an arc in ∂Δ properly embedded in Di, Δ∩∂+C=β is an arc in ∂Δ, and α∩β=∂α=∂β, α∪β=∂Δ. α cuts Di into two disks Di1 and Di2. Set D′i=Di1∪Δ and D″i=Di2∪Δ, and move D′i by a small isotopy such that D′i∩D″i=∅. Set D′=(D∖{Di})∪{D′i,D″i}. It is clear that D′ is also a defining disk system for C. We say that D′ is a slide of D along Δ.
Let Mi be a compact connected 3-manifold, Fi⊂∂Mi a connected surface such that no component of ∂Fi bounds a disk in ∂Mi, i=1,2, and h:F1→F2 a homeomorphism. Set M=M1∪hM2, F1=F=F2 in M, and call M an amalgamation of M1 and M2 along F. M is also denoted as M1∪FM2, and F is called a splitting surface of M.
Clearly, if F is a disk, M is a boundary connected sum of M1 and M2, and is also denoted by M1#∂M2; if both M1 and M2 are compression bodies and ∂+M1=F=∂+M2, then M1∪FM2 is a Heegaard splitting for M, and F is a Heegaard surface of M; if both M1 and M2 are handlebodies, then M1∪FM2 is called an H′-splitting for M, and F is an H′-splitting surface of M (here, F is not necessarily closed).
It is well known that any compact connected orientable 3-manifold admits a Heegaard splitting[8]. It has been shown in [5] that any compact connected orientable 3-manifold admits an H′-splitting.
For M=M1∪FM2, suppose that F is connected and compressible in both M1 and M2. Let Di be a compressing disk of F in Mi, i=1,2. We say that F is stabilized if |∂D1∩∂D2|=1; reducible if ∂D1=∂D2; and weakly reducible if ∂D1∩∂D2=∅. Otherwise, F is unstabilized, irreducible, or strongly irreducible, respectively.
Let V∪FW be a Heegaard splitting of genus g for M, and T∪TT′ the Heegaard splitting of genus 1 for S3. The connected sum (V∪FW)#(T∪TT′) is a Heegaard splitting of genus g+1 for M, and is called an elementary stabilization of V∪FW. A Heegaard splitting V′∪F′W′ is called a stabilization of V∪FW if it is obtained by a finite number of elementary stabilization from V∪FW.
In the following, we collect some known facts which will be used in Section 3.
Lemma 2.1. [10] Let H be a handlebody of genus n≥2 and F be an incompressible surface in H. Then, the manifold obtained by cutting H open along F is a union of handlebodies.
Lemma 2.2. [10] Let A be a spanning annulus in the compression body C. Then, there exists a defining system D of disks for C such that A is disjoint from any disk in D.
The following is the uniqueness theorem of Heegaard splittings for S3, due to Waldhausen [11].
Theorem 2.3. Any positive genus Heegaard splitting of S3 is stabilized.
A handlebody H has a natural Heegaard splitting: A surface F in int(H) which is parallel to ∂H splits H into a handlebody (≅H) and a trivial compression body. Call it the trivial splitting of H. A Heegaard splitting of H is called standard if it is a stabilization of the trivial splitting.
Two consequences of Theorem 2.3 are as follows.
Theorem 2.4. [12] Any Heegaard splitting of a handlebody H of positive genus is standard.
Theorem 2.5. [10] Let M be an irreducible 3-manifold and X∪FY be a Heegaard splitting for M. Suppose that X∪FY is reducible. Then, X∪FY is stabilized.
Let H be a handlebody and J be a collection of pairwise disjoint s.c.c. in ∂H. Denote by H(J) the 3-manifold obtained by attaching 2-handles to H along the curves in J.
Definition 2.6. Let J={J1,⋯,Jm} be a collection of simple closed curves in the boundary of a handlebody H of genus n.
(1) If {[J1],⋯,[Jm]}⊂π1(H) (after some conjugation) can be extended to a basis of π1(H), then we say that J is primitive in H.
(2) If the curves in J are pairwise disjoint, and H(J) is a handlebody (of genus n−m), then we say that J is quasi-primitive in H.
In the Definition 2.6, if the curves in J are pairwise disjoint, it is clear that J is primitive implies that J is quasi-primitive. But the converse is generally not true except for m=1. It is a theorem in [13] that if all subsets J′ of J are quasi-primitive in H, then J is primitive in H.
The following theorem gives a characteristic of an H′-splitting X∪FY for a handlebody H, where F is incompressible in H.
Theorem 2.7 ([6]). Let X∪FY be an H′-splitting for a 3-manifold M, where g(X),g(Y)≥2. Suppose that F is incompressible in both X and Y. Then, M is a handlebody if and only if there exists a basis curve set J={J1,⋯,Jm} of π1(F) with a partition (J1,J2) of J such that J1 is primitive in X and J2 is primitive in Y.
A properly embedded annulus A in a compression body C is called a spanning annulus if A is incompressible in C and the two components of ∂A are lying in ∂+C and ∂−C respectively.
The following is a well-known fact, refer to [10] for a proof.
Theorem 2.8. Let V be a compression body with ∂−V≠∅ and F be an incompressible, ∂-incompressible surface properly embedded in M. Then, each component of F is either a spanning annulus, an essential disk, or parallel to a component of ∂−V in V.
Let Pb be a connected planar surface with b boundary components, b>0. Let Γ be a collection of pairwise disjoint and non-parallel simple arcs properly embedded in Pb, such that the surface obtained by cutting Pb along Γ is a union of m disks, it is clear that m≤b−1.
Haken's lemma states that any Heegaard splitting of a reducible 3-manifold is reducible. It was generalized to the ∂-reducible 3-manifolds as follows: For any Heegaard splitting V∪SW of a ∂-reducible 3-manifold M, there exists a compression disk D for ∂M in M, such that D∩S is a single circle. The following is a stronger version of this result, and an outline proof is included for convenience.
Lemma 2.9. Let M be a connected 3-manifold with a Heegaard splitting V∪SW and F be a sub-surface of ∂−V. Suppose that F is incompressible in V and compressible in M. Then, there exists a compresing disk D for F in M, such that D∩S is a single circle.
Proof. By assumption, F⊂∂−V is incompressible in V and compressible in M, so V is non-trivial. Any compressing disk E of F in M intersects S non-empty. Let Γ be a spine of W such that Γ is in general position with E∩W. Thus E∩W consists of pairwise disjoint disks in W. Choose such a compressing disk D of F in M, such that |D∩S| is minimal among all such disks in M. Set P=D∩V. P is a connected planar surface. We may further assume that P is incompressible in V. If P is an annulus, then the lemma holds.
Assume m=|S∩P|≥2. By Theorem 2.8, P is ∂-compressible in V. Let Δ be a ∂-compresing disk for P. Push D along Δ by isotopy to get D1, and denote D1∩V by P1. Then, P1 is the surface obtained by doing the boundary compressing P in V along Δ. If P1 is still ∂-compressible in V, do the similar operation as above. After a finite number such operations, we can isotope D to D′ in M such that D′∩V consists of a spanning annulus and a collection of essential disks in V. It is easy to see m′=|P′∩S|≤m. Thus, Q′=D′∩W is a connected planar surface properly embedded in W with ∂Q′⊂S.
We may further assume that Q′ is incompressible in M. Similarly, we can isotope D′ to D″ in M such that Q″=D″∩W consists of a collection of essential disks in W with m″=|Q″∩S|<|Q′∩S|≤m, a contradiction to the minimality of |D∩S|.
A similar result to Theorem 2.5 for a reducible H′-splitting of an irreducible 3-manifold holds as follows.
Theorem 3.1. Let M be an irreducible 3-manifold, and X∪FY an H′-splitting for M. Suppose that X∪FY is reducible. Then X∪FY is stabilized.
Proof. By assumption, X∪FY is reducible. There exists an essential circle α in F, and α bounds a disk D in X and a disk E in Y, respectively. We divide it into three cases to discuss.
Case 1. α is non-separating in F. Then, there exists a s.c.c. β in F such that β meets α in one point. Let N be a regular neighborhood of D∪E∪β in M. Then, N is a once-punctured S2×S1, contradicting to that M is irreducible. See Figure 1 (a) below. Thus, Case 1 cannot happen.
Case 2. α is separating in F, and cuts F into two surfaces F1 and F2 with (∂Fi)∩∂M≠∅, i=1,2. In the case, D∪E is a separating 2-sphere in M, which cuts M into M1 and M2, and neither M1 nor M2 is a 3-ball, again contradicting to that M is irreducible. See Figure 1 (b) below. Thus, Case 2 cannot happen.
Case 3. α is separating in F, and cuts out of a once-punctured surface F′ of positive genus from F with ∂F′=α. In the case, D cuts out of a handlebody H1 from X and E cuts out of a handlebody H2 from Y, H1∩F=F′=H2∩F. Set M′=H1∪F′H2, then ∂M′=D∪E. By the irreducibility of M, M′ is a 3-ball. By capping of M′ with a 3-ball B3, we get M″=M′∪∂B3≅S3, and F′ naturally extended to a Heegaard surface of positive genus for M″. By Theorem 2.3, such a Heegaard splitting is stabilized. It follows that X∪FY is stabilized.
The following is a direct consequence of Theorem 3.1:
Corollary 3.2. Let H be a handlebody of genus g≥1. If X∪FY is a reducible H′-splitting for H, then X∪FY is stabilized.
Let H be a handlebody of genus g and J be an s.c.c. on ∂H. If there exists a disk D properly embedded in H with |J∩∂D|=1, we call J a longitude of H. It is clear that X#∂Y is a handlebody if and only if both X and Y are handlebodies. For an H′-splitting X∪AY for a 3-manifold M, where A is an annulus, it is known (refer to [14] for a proof) that M is a handlebody if and only if the core curve of A is a longitude of either X or Y. In particular, if HJ is the manifold obtained by adding a 2-handle to the handlebody along an s.c.c. J on ∂H, then HJ is a handlebody if and only if J is a longitude of H (or equivalently, J is primitive in H).
Theorem 3.3. Let H be a handlebody of genus g≥1, X∪FY an H′-splitting for H. Suppose that F is weakly reducible in H. Then either X∪FY is stabilized, or there exists a collection D={D1,⋯,Dm} (E={E1,⋯,En}, resp.) of pairwise disjoint compressing disks of F in X (in Y, resp.), such that ∂D∩∂E=∅, and if we denote by F′ the surface obtained by compressing F in H along D∪E, then F′ is incompressible in H. Moreover, if D′ (E′, resp.) is the subset of D (E, resp.) which consists of only the non-separating disks in X (in Y, resp.), then ∂D′ is quasi-primitive in Y and ∂E′ is quasi-primitive in X.
Proof. By assumption, F is weakly reducible in H. There exist compresing disks D1⊂X and E1⊂Y of F with ∂D1∩∂E1=∅. If ∂D1 is parallel to ∂E1, then X∪FY is reducible (therefore, stabilized, by Corollary 3.2). Extend D1 and E1 to a collection D={D1,⋯,Dm} (E={E1,⋯,En}, resp.) of pairwise disjoint compressing disks of F in X (in Y, resp.) in such a way that ∂D∩∂E=∅, and the following conditions are satisfied:
(1) If we denote by FX (FY, resp.) the surface obtained from F by compressing X (Y, resp.) along D (E, resp.), then no component of FX (FY, resp.) whose boundary is a subset of the set of the cutting sections of D (E, resp.) is a planar surface.
(2) If Δ is a compresing disk of F in X (Y, resp.) with ∂Δ∩(∂D∪∂E)=∅, then ∂Δ cuts out of a planar surface P from FX (FY, resp.) with ∂P−{∂Δ}⊂∂D (∂P−{∂Δ}⊂∂E, resp.).
(3) If F′ is the surface obtained from F by compressing F along D∪E, C(F′)=−χ(F′), the comlexity of F′, is minimal over all such D∪E.
By (1) and (2), each ∂Di (∂Ej, resp.) either is non-separating in F, or cuts F into two pieces, each of which intersects ∂H non-empty.
Denote by ˜F the surface obtained by cutting F open along ∂D∪∂E. If ˜F has a planar component Q with ∂Q=L1∪L2, where L1 is a subset of the set of the cutting sections of ∂D and L2 is a subset of the set of the cutting sections of ∂E, then let α be an s.c.c. on Q such that α cuts Q into Q1 and Q2 with ∂Q1−α=L1 and ∂Q2−α=L2. Therefore, by above (1), L1,L2≠∅. Thus, α is essential in F and α bounds disks in both X and Y. Hence, X∪FY is reducible (therefore, stabilized, by Corollary 3.2). In the following, we assume that ˜F has no such planar component.
Let N1 (N2, resp.) be a regular neighborhood of D in X (E in Y, resp.) such that (N1∩F)⋂(N2∩F)=∅. Set X∗=¯X∖N1, Y∗=¯Y∖N2. X∗ (Y∗, resp.) is in fact the manifold obtained by compressing X along D (Y along E, resp.), therefore it is a union of handlebodies. Assume that after compressing X along D (Y along E, resp.), F is changed into F1 (F2, resp.). Since F is separating in H, each component of F1 (F2, resp.) is separating in H. If F1 (F2, resp.) has a component which is a closed surface of positive genus, it bounds a handlebody in X∗ (Y∗, resp.), contradicting to the choice of D (E, resp.). Thus, each component of F1 (F2, resp.) has non-empty boundary.
Set X′=X∗∪N2, Y′=Y∗∪N1, and F′=X′∩Y′. Then, H=X′∪F′Y′. By above (1), no component of ∂X′ (∂Y′, resp.) is a 2-sphere. Suppose that F′ has k components F′1,⋯,F′k. It follows that each F′i is separating in H, 1≤i≤k. With no loss, assume that F′ cuts H into k+1 pieces M1,⋯,Mk+1, and
H=((M1∪F′1M2)∪F′2⋯)∪F′kMk+1, |
where each Mi is the 3-manifold obtained by adding some 2-handles (possibly empty) to a handlebody in X∗ or Y∗, 1≤i≤k+1.
We now show that F′ is incompressible in H. Otherwise, some F′i is compressible in Mi. Without loss of generality, assume that Mi is a 3-manifold obtained by adding a subset N′ (2-handles) of N2 to a handlebody H∗ in X∗ along the curves on a component S′ of FX. Say N′={η(Ei1),...,η(Eis)}. Let S be a surface in the interior of H∗ which is parallel to ∂H∗. Then, S is a Heegaard surface in Mi. In fact, S splits Mi into a handlebody V≅H∗ and a compression body W, and E′={Ei1,...,Eis} can be extended to a defining disk system for W. Note that F′i⊂∂−W. By Lemma 2.9, there exists a compressing disk D of F′i in Mi, such that D∩S is an essential circle in S. Set A=D∩W, A is a spanning annulus in W. See Figure 2 below. By Lemma 2.2, there exists a defining disk system E∗ for W, such that A is disjoint from each disk in E∗. Assume that A and E′ are in general position. By an innermost argument, we may further assume that Λ=A∩⋃sj=1Eij has no circle components. Let γ be an arc component of Λ which is outermost in A. Then, γ cuts out of a disk Δ from A with int(Δ)∩⋃sj=1Eij=∅. Slide E′ along Δ to a new defining disk system for W with less intersection with A. After finite such slides, we can obtain the defining disk system E∗ for W which is disjoint from A. Since both the component of ∂A lying in S and ∂E′ are lying in the parallel copy S″ of S′ in S, it follows that ∂E∗⊂S″. It is clear that there is a subset ˜E of E∗, such that ˜E is a defining disk system for W and ˜E contains s disks. Set D1=D∪{D}, and E1=(E∖E′)⋃(˜E∩Y). Note that ∂D is essential in F′. Denote by F″ the surface obtained from F by compressing F in H along D1∪E1, it is clear that C(F″)<C(F′), a contradiction to the minimality of C(F′).
Thus, F′ is incompressible in H. It follows from Lemma 2.1 that each Mi is a handlebody, 1≤i≤k+1. If D′ (E′, resp.) is the subset of D (E, resp.) which consists of only the non-separating disks in X (in Y, resp.), it follows that ∂D′ is quasi-primitive in Y and ∂E′ is quasi-primitive in X.
This completes the proof.
By a similar arguments to the proof of Theorem 3.3, we have the following theorem. The proof is omitted.
Theorem 3.4. Let H be a handlebody of genus g≥1, X∪FY an H′-splitting for H. Suppose that F is compressible in X (or Y) and incompressible in Y (or X), or F is compressible in both X and Y, and is strongly irreducible in H. Then, there exists a collection D={D1,⋯,Dm} (E={E1,⋯,En}, resp.) of pairwise disjoint compressing disks of F in X (in Y, resp.), such that if we denote by F′ the surface obtained by compressing F in H along D (E, resp.), then F′ is incompressible in H. Moreover, if D′ (E′, resp.) is the subset of D (E, resp.) which consists of only the non-separating disks in X (in Y, resp.), then ∂D′ is quasi-primitive in Y (∂E′ is quasi-primitive in X, resp.).
Use the notations as in Theorems 3.3 and 3.4. We call ∂D′∪∂E′ a reducing system of F (in Theorem 3.4, ∂D′ or ∂E′=∅). Combining Theorem 3.3, Theorem 3.4, and Theorem 2.7, we have the following direct corollary.
Theorem 3.5. Let H be a handlebody of genus g≥1, X∪FY an H′-splitting for H. Then, either X∪FY is stabilized, or there exists a reducing system J1∪K1 of F, such that J1 is quasi-primitive in Y, and K1 is quasi-primitive in X. Moreover, if the incompressible surface F′ obtained by compressing F in H along the disks in H bounded by J1∪K1 is connected, then there exists a basis curve set L of π1(F′) with a partition (J2,K2) of L such that J2 is primitive in X and K2 is primitive in Y.
We remark that a similar conclusion as in Theorem 3.5 holds when F′ not connected. We omit the statement.
We describe a characteristic of an H′-splitting X∪FY to denote a handlebody H, where F may be compressible in H. This generalizes an earlier result (Theorem 2.7) in which the H′-surface in a handlebody is assumed to be incompressible.
Yan Xu: Writing the original draft, Investigation, Methodology; Bing Fang: Writing-review & editing the draft, Investigation; Fengchun Lei: Conceptualization, Methodology, Supervision, Writing-review & editing the draft. All authors agreed to publish the final version of the manuscript.
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
This work was supported by a grant (No.12071051) of NSFC.
The authors are grateful to the referees for their insightful comments and suggestions, which helped to improve the papers presentation.
The authors declare no conflict of interest.
[1] |
J. S. Downing, Decomposing compact 3-manifolds into homeomorphic handlebodies, Proc. Amer. Math. Soc., 24 (1970), 241–244. https://doi.org/10.1090/S0002-9939-1970-0250318-1 doi: 10.1090/S0002-9939-1970-0250318-1
![]() |
[2] |
L. G. Roeling, The genus of an orientable 3-manifold with connected boundary, Illinois J. Math., 17 (1973), 558–562. https://doi.org/10.1215/ijm/1256051475 doi: 10.1215/ijm/1256051475
![]() |
[3] |
S. Suzuki, Handlebody splittings of compact 3-manifolds with boundary, Rev. Mat. Complut., 20 (2007), 123–137. http://dx.doi.org/10.5209/rev_REMA.2007.v20.n1.16548 doi: 10.5209/rev_REMA.2007.v20.n1.16548
![]() |
[4] |
A. Casson, C. Gordon, Reducing heegaard splittings, Topol. Appl., 27 (1987), 275–283. https://doi.org/10.1016/0166-8641(87)90092-7 doi: 10.1016/0166-8641(87)90092-7
![]() |
[5] |
Y. Gao, F. Li, L. Liang, F. Lei, Weakly reducible H′-splittings of 3-manifolds, J. Knot Theor. Ramif., 30 (2021), 2140004. https://doi.org/10.1142/S0218216521400046 doi: 10.1142/S0218216521400046
![]() |
[6] |
F. Lei, H. Liu, F. Li, A. Vesnin, A necessary and sufficient condition for a surface sum of two handlebodies to be a handlebody, Sci. China Math., 63 (2020), 1997–2004. https://doi.org/10.1007/s11425-019-1647-9 doi: 10.1007/s11425-019-1647-9
![]() |
[7] | A. Hatcher, Notes on basic 3-Manifold topology, 2000. Available from: https://api.semanticscholar.org/CorpusID: 9792594 |
[8] | W. Jaco, Lectures on three manifold topology, Providence: American Mathematical Soc., 1980. http://dx.doi.org/10.1090/cbms/043 |
[9] | M. Scharlemann, Heegaard splittings of compact 3-manifolds, arXiv Prepr. Math., 2000. Available from: https://arXiv.org/pdf/math/0007144 |
[10] | J. Johnson, Notes on Heegaard splittings, Preprint, 2006. Available from: http://pantheon.yale.edu/jj327/ |
[11] |
F. Waldhausen, Heegaard-Zerlegungen der 3-Sph¨are, Topology, 7 (1968), 195–203. https://doi.org/10.1016/0040-9383(68)90027-X doi: 10.1016/0040-9383(68)90027-X
![]() |
[12] |
M. Scharlemann, A. Thompson, Heegaard splittings of (surface)×I is standard, Math. Ann., 295 (1993), 549–564. https://doi.org/10.1007/BF01444902 doi: 10.1007/BF01444902
![]() |
[13] |
C. M. Gordon, On primitive sets of loops in the boundary of a handlebody, Topol. Appl., 27 (1987), 285–299. https://doi.org/10.1016/0166-8641(87)90093-9 doi: 10.1016/0166-8641(87)90093-9
![]() |
[14] |
F. Lei, Some properties of an annulus sum of 3-manifolds, Northeast. Math. J., 10 (1994), 325–329. https://doi.org/10.13447/j.1674-5647.1994.03.007 doi: 10.13447/j.1674-5647.1994.03.007
![]() |