The Quarterly Journal of Mathematics, Volume Advance Article (2) – Oct 10, 2017

18 pages

/lp/ou_press/complexities-of-3-manifolds-from-triangulations-heegard-splittings-and-c3JSyQcgFn

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- Oxford University Press
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- 0033-5606
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- 1464-3847
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- 10.1093/qmath/hax041
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Abstract We study complexities of 3-manifolds defined from triangulations, Heegaard splittings, and surgery presentations. We show that these complexities are related by linear inequalities, by presenting explicit geometric constructions. We also show that our linear inequalities are asymptotically optimal. Our results are used in another paper of the author to estimate Cheeger-Gromov L2ρ-invariants in terms of geometric group theoretic and knot theoretic data. 1. Introduction and main results In this paper, we study the relationship between various notions of complexities of 3-manifolds. In what follows, we always assume that 3-manifolds are compact. 1.1. Simplicial complexity The first notion of complexity we consider is defined from triangulations. In this paper, a triangulation designates a simplicial complex structure. Definition 1.1 For a 3-manifold M, the simplicial complexity csimp(M) is defined to be the minimal number of 3-simplices in a triangulation of M. A similar notion of complexity defined from more flexible triangulations is often considered in the literature (for example, see [7–9, 16]): a pseudo-simplicial triangulation of a 3-manifold M is defined to be a collection of 3-simplices together with affine identifications of faces from which M is obtained as the quotient space. The pseudo-simplicial complexity, or the complexity c(M) of M is defined to be the minimal number of 3-simplices in a pseudo-simplicial triangulation. For closed irreducible 3-manifolds, c(M) agrees with Matveev’s complexity [13] defined in terms of spines, unless M=S3, RP3 or L(3,1). Since the second barycentric subdivision of a pseudo-simplicial triangulation is a triangulation and a 3-simplex is decomposed to (4!)2=576 3-simplices in the second barycentric subdivision, we have 1576·csimp(M)≤c(M)≤csimp(M). 1.2. Heegaard–Lickorish complexity Recall that a Heegaard splitting of a closed 3-manifold is represented by a mapping class in the mapping class group Mod(Σg) of a surface Σg of genus g. (Our precise convention is described in the beginning of Section 3. The identity mapping class gives the standard Heegaard splitting of S3 as shown in Fig. 1.) It is well known that Mod(Σg) is finitely generated; Lickorish showed that Mod(Σg) is generated by the ±1 Dehn twists about the 3g−1 curves αi, βi and γi shown in Fig. 1 [10, 11]. Figure 1. View largeDownload slide Standard Dehn twist curves of Lickorish. Figure 1. View largeDownload slide Standard Dehn twist curves of Lickorish. From this, a geometric group theoretic notion of complexity is defined for 3-manifolds as follows. Definition 1.2 The Heegaard–Lickorish complexity cHL(M) of a closed 3-manifold M is defined to be the minimal word length, with respect to the Lickorish generators, of a mapping class h∈Mod(Σg) on a surface Σg of arbitrary genus which gives a Heegaard splitting of M. Note that both the genus g of a Heegaard surface Σg and the mapping class h vary in taking the minimum in Definition 1.2. By definition, cHL(S3)=0. We remark that the Heegaard–Lickorish complexity tells us more delicate information than the Heegaard genus. It turns out that the difference of the Heegaard–Lickorish complexities of two 3-manifolds with the same Heegaard genus can be arbitrarily large, whereas the Heegaard genus of a 3-manifold is bounded by twice its Heegaard–Lickorish complexity. See Lemma 3.1 and related discussions in Section 3. Our first result is the following relationship between the two complexities defined above. Theorem A For any closed 3-manifold M≠S3, csimp(M)≤552·cHL(M). We remark that upper bounds to (pseudo-)simplicial complexity in terms of a Heegaard splitting were studied earlier in the literature, for instance see [13, Proposition 3] and [15, Proposition 2.1.8]. In many cases, Theorem A provides a sharper upper bound. For more about this, see Remark 3.5 as well as Theorems C and D below which concern the optimality of our bound. The optimality is essential in an application of [4] (see the last part of the introduction). 1.3. Surgery complexity To define another notion of complexity of 3-manifolds from knot theoretic information, we consider Dehn surgery with integral coefficients. For a framed link L in S3, let f(L)=∑i∣fi(L)∣, where fi(L)∈Z is the framing on the ith component of L. If a component K of L is contained in an embedded 3-ball in S3 which is disjoint from other components, then we call K a split component. Let n(L) be the number of split unknotted zero framed components of L. An example with f(L)=2, n(L)=1 is illustrated in Fig. 2. We denote by c(L) the crossing number of a link L in S3, that is, c(L) is the minimal number of crossings in a planar diagram of L. As a convention, if L is empty, then c(L)=f(L)=n(L)=0. Definition 1.3 The surgery complexity of a closed 3-manifold M is defined by csurg(M)=minL{2c(L)+f(L)+n(L)}, where L varies over framed links in S3 from which M is obtained by surgery. Figure 2. View largeDownload slide A framed link L with f(L)=2, n(L)=1. Figure 2. View largeDownload slide A framed link L with f(L)=2, n(L)=1. We remark that we bring in n(L) to detect S1×S2 summands, which can be added to any 3-manifold by connected sum without altering c(L) and f(L) of a framed link L giving the 3-manifold. Note that n(L)=0 for any L that gives M if M has no S1×S2 summand. In particular, it is the case if M is irreducible. Note that csurg(S3)=0 by our convention. Our second result is the following relationship between the simplicial complexity and the surgery complexity. Theorem B For any closed 3-manifold M≠S3, csimp(M)≤72·csurg(M). We remark that Matveev gave a similar inequality which relates the complexity c(M) to a surgery presentation [15, Proposition 2.1.13]. The proofs of Theorems A and B consist of geometric arguments which explicitly construct triangulations from Heegaard splittings and from surgery presentations. Details are given in Sections 2 and 3. 1.4. Optimality of Theorems A and B It is natural to ask how sharp the inequalities in Theorems A and B are. This seems to be a non-trivial problem, since it appears to be hard to determine the complexities we consider, or even to find an efficient lower bound for them. We remark that the determination and lower bound problems for the pseudo-simplicial complexity c(M) have been studied extensively in the literature and regarded as difficult problems [9, 14]. We show that the linear inequalities in Theorems A and B are asymptotically optimal. This can be described in terms of standard notations for asymptotic growth, as follows. Recall that we write f(n)∈O(g(n)) if f is bounded above by g asymptotically, that is, limsupn→∞∣f(n)/g(n)∣ is finite. Also, f(n)∈o(g(n)) if f(n) is dominated by g(n) asymptotically, that is, limsupn→∞∣f(n)/g(n)∣=0. We write f(n)∈Ω(g(n)) if f(n) is not dominated by g(n). Define two functions sHL(ℓ) and ssurg(k) by sHL(ℓ)=sup{csimp(M)∣cHL(M)≤ℓ},ssurg(k)=sup{csimp(M)∣csurg(M)≤k}, where the supremums exist by Theorems A and B. In other words, sHL(ℓ) is the ‘largest possible value’ of the simplicial complexity for 3-manifolds with Heegaard–Lickorish complexity ℓ or less. We can interpret ssurg(k) similarly. Theorem C sHL(ℓ)∈O(ℓ)∩Ω(ℓ)and ssurg(k)∈O(k)∩Ω(k). As explicit examples, the lens spaces L(n,1) satisfy the following: Theorem D For any n>3, 14 357 080·cHL(L(n,1))≤csimp(L(n,1)),14 357 080·csurg(L(n,1))≤csimp(L(n,1)). We also prove a similar inequality for a larger class of 3-manifolds. See Theorem 4.4 and related discussions in Section 4. The proofs of Theorems C and D are given in Section 4. 1.5. Applications to universal bounds for Cheeger–Gromov invariants Results in this paper are closely related to the recent development of a topological approach to the universal bounds of Cheeger–Gromov L2ρ-invariants in [4]. In fact, Theorems A and B of this paper are used as essential ingredients in [4] to give explicit linear estimates of Cheeger–Gromov ρ-invariants of 3-manifolds in terms of geometric group theoretical and knot theoretical data. See [4, Theorems 1.8 and 1.9]. This application is a major motivation of the present paper. Our inequalities in Theorems A and B are sharp enough, compared with earlier similar work, to give results that the linear estimates in [4] are asymptotically optimal. See [4, Theorem 7.8]. On the other hand, the lower bounds in Theorem D are proven by employing results of [4] which relate triangulations and the Cheeger–Gromov ρ-invariants. See Section 4 for more details. 2. Linear complexity triangulations from surgery presentations In this section, we present a construction of a triangulation from a surgery presentation. Lemma 2.1 Suppose L is a framed link in S3. Suppose there is a planar diagram D with c or fewer crossings for L, in which there is no local kinkand each zero framed component of L is involved in a crossing. Let wi∈Zbe the writhe of the ith component in the diagram D. Then, the 3-manifold M obtained by surgery on L has simplicial complexity at most 96c+48∑∣fi(L)−wi∣. Example 2.2 Consider the stevedore knot, which is 61 in the table in Rolfsen [17], or KnotInfo [5]. It has a planar diagram with six crossings, where two of them have the same sign, but the other four have the opposite sign. It follows that the zero surgery manifold M of 61 satisfies csimp(M)≤96·6+48·2=672. Before we prove Lemma 2.1, we prove Theorem B using Lemma 2.1. Proof of Theorem B Recall that Theorem B says csimp(M)≤72·csurg(M) for M≠S3. We need the following two observations: first, we have csimp(M1#M2)≤csimp(M1)+csimp(M2)−2, (2.1) since the connected sum of two triangulated 3-manifolds can be performed by deleting a 3-simplex from each and then glueing faces. Secondly, we have csimp(S1×S2)≤72. (2.2) For instance, by taking the product of a triangle triangulation of S1 and its suspension which is a triangulation of S2, and then by applying the standard prism decomposition to each product Δ1×Δ2 (see Fig. 3), we obtain a triangulation of S1×S2 with 3·6·3=54 tetrahedra. Choose a framed link L such that M is obtained by surgery on L and 2c(L)+f(L)+n(L)=csurg(M). Choose a planar diagram D for L with a minimal number of crossings, and let D0 be the subdiagram of D consisting of zero-framed split components of L. From the minimality, it follows that D has no local kink and D0 consists of zero-framed circles with no crossing. Also every zero-framed component of L in D−D0 is involved in a crossing. Let M′ be the 3-manifold obtained by surgery along the given framing of D−D0. Since a component of D0 contributes an S1×S2 summand, M=M′#(n(L)·(S1×S2)). If D=D0, then c(L)=f(L)=0 and M′=S3; also, n(L)≥1 since M≠S3. It follows that csimp(M)≤n(L)·csimp(S1×S2)≤72·n(L) by using (2.1) and (2.2). This is the desired conclusion for this case. If D≠D0, we have csimp(M)≤csimp(M′)+n(L)·csimp(S1×S2)≤csimp(M′)+72·n(L) (2.3) by using (2.1) and (2.2). The number of crossings of D−D0 is equal to that of D, which is equal to c(L) by our choice of D. Let wi be the writhe of the ith component in D−D0, and fi be its given framing. Since a crossing in the diagram contributes 1, 0 or −1 to wi for some i, it follows that ∑∣wi∣≤c(L). Therefore, we have csimp(M′)≤96·c(L)+48·∑∣fi−wi∣≤96·c(L)+48·(f(L)+c(L))≤72·(2c(L)+f(L)) (2.4) by Lemma 2.1. From (2.3) and (2.4), the desired conclusion follows.□ Proof of Lemma 2.1 We will construct a triangulation of the exterior of L which is motivated from J. Weeks’ SnapPea (see [18]), and then will triangulate the Dehn filling tori in a compatible way. In what follows we view D as a planar diagram lying on S2. By the subadditivity (2.1), we may assume that the diagram D is non-split, that is, any simple closed curve in S2 disjoint from D bounds a disk disjoint from D. This is equivalent to that every region of D is a disk. Either D has at least one crossing, or D is a circle with no crossings. First, suppose that it is the former case. Consider the dual graph G0 of D, whose regions are quadrangles corresponding to crossings. (Since D has no local kinks, the four vertices of each quadrangles are mutually distinct.) For each component of the link L, choose an edge of G0 which is dual to the component (that is, the edge intersects a strand of D that belongs to the component), and add 2∣fi−wi∣ parallels of the edge, where fi=fi(L) is the framing and wi∈Z is the writhe of the component. Denote the resulting graph by G. For an example, see the left of Fig. 4, which illustrates the case of a (+1)-framed figure eight. View the link L as a submanifold of S2×[−1,1] which projects to D under S2×[−1,1]→S2, and remove from S2×[−1,1] an open tubular neighborhood ν(L) of L, which is tangential to S2×{−1,1} at (eachcrossing)×{−1,1}; cutting along G×[−1,1], we obtain pieces of two types: (i) cubes with two tunnels, which correspond to the crossings of D, and (ii) those of the form (2-gon)×[−1,1] with a tunnel removed, which correspond to the edges of G−G0. See the middle of Fig. 4. Cut each piece along D×[−1,1]. In case of type (i), we obtain four equivalent subpieces. See the top right of Fig. 4. The hatched quadrangles represent ∂ν(L). Each of the four subpieces can be viewed as a cube shown in the left of Fig. 5. Let p be the vertex shown in Fig. 5, and triangulate the three square faces not adjacent to p as in the left of Fig. 5. By taking a cone from p, we obtain a triangulation of the each type (i) subpiece. Since the triangulation of the faces away from p has 14 triangles, the cone triangulation of a type (i) subpiece has 14 tetrahedra. In case of type (ii), by cutting each piece along D×[−1,1], we obtain two subpieces, each of which are as in the bottom right of Fig. 4. For each type (ii) subpiece, triangulate the front face as shown in the right of Fig. 5, and then triangulate the subpiece by taking the cone of the union of the front face and top triangle from the vertex q, similar to the above type (i) case. This triangulation of a type (ii) subpiece has seven tetrahedra. Suppose D has c crossings. For brevity, denote δ≔∑∣fi−wi∣. There are 4c subpieces of type (i) and 4δ subpieces of type (ii). By applying the above to each of them, we obtain a triangulation of S2×[−1,1]⧹ν(L), which has 14·4c+7·4δ=56c+28δ tetrahedra. For t=±1, the triangulation restricts to a triangulation of S2×{t} with 8c+4δ triangles, since the top of type (i) and (ii) subpieces consists of two triangles and a single triangle, respectively. Attaching two 3-balls triangulated as the cone of these triangulations, we obtain a triangulation of S3⧹ν(L) which has (56c+28δ)+2·(8c+4δ)=72c+36δ tetrahedra. In our triangulation, there are 8c+4δ hatched quadrangular regions, and they are paired up to form 4c+2δ annuli, and the ith boundary component of ν(L) is a union of 2ki+2∣fi−wi∣ such annuli, where ki is the number of times the ith component of L passes through a crossing. We have ∑ki=2c. (Since a component may pass through the same crossing twice, ki may not be equal to the number of crossings that the component passes through.) See the left of Fig. 6; the hatched meridional annulus is one of these 2ki+2∣fi−wi∣ annuli. On the ith boundary component of ν(L), take the top and bottom edges of the hatched quadrangles in type (i) subpieces, and the diagonal edges used to triangulate the hatched quadrangles in type (ii) subpieces. We may assume that the union of these edges consists of two parallel circles, say αi and αi′, by appropriately altering the choices of diagonals used above to triangulate the hatched quadrangles if necessary. See the left of Fig. 6 in which αi and αi′ are shown as thick curves. Moreover, we may assume that the framing represented by αi differs from the blackboard framing by fi−wi; that is, whenever αi passes through a type (ii) piece, a half twist with the same sign as that of fi−wi is introduced with respect to the blackboard framing, while αi runs along the blackboard framing in type (i) pieces. See the left of Fig. 6, which illustrates the case of fi−wi=1. Since the blackboard framing is equal to wi, it follows that αi represents the given framing fi. Take a solid torus D2×S1 for each component of L. Attach the solid tori to the exterior S3⧹ν(L) along orientation reversing homeomorphisms of boundary tori, which takes the curves αi and αi′ to meridians bounding disks and takes a hatched annulus to a longitudinal annulus, as shown in Fig. 6. Pulling back the triangulation of ∂(S3⧹ν(L)), we obtain a triangulation of ∂(D2×S1). It extends to a triangulation of D2×S1 as follows. By cutting the D2×S1 along the meridional disks bounded by αi and αi′, we obtain two solid cylinders D2×[0,1]. Note that we already have 2ki+2∣fi−wi∣ vertices on ∂D2×0. We triangulate D2×0 into 2ki+2∣fi−wi∣ triangles, by drawing edges joining the vertices to the center of D2×0. See the bottom of Fig. 6. Taking the product with [0,1], we decompose D2×[0,1] into 2ki+2∣fi−wi∣ triangular prisms. Note that each prism corresponds to a hatched quadrangle. Finally, we apply the standard prism decomposition (Fig. 3) to each prism. Since each prism gives three tetrahedra and there are 8c+4δ hatched quadrangles, the union of all the Dehn filling solid tori is decomposed into 3(8c+4δ)=24c+12δ tetrahedra. The triangulation of our surgery manifold M is obtained by adjoining the Dehn filling tori triangulations to that of the exterior. By the above tetrahedra counting, it follows that the number of tetrahedra in M is at most (72c+36δ)+(24c+12δ)=96c+48δ. This completes the proof when there is at least one crossing in D. Now, suppose D consists of a single circle without crossings. Note that the writhe is zero in this case. Let f1∈Z be the given framing. By the hypothesis, f1≠0. We need to prove that csimp(M)≤48∣f1∣. If f1=±1, then M=L(f1,1)=S3, and it is straightforward to verify that csimp(S3)≤48. (For instance, triangulate the equator S2⊂S3 into four triangles, by viewing it as the boundary of a 3-simplex, and triangulate the upper and lower hemispheres by taking a cone of the equator, to obtain a triangulation of S3 with eight tetrahedra.) Suppose ∣f1∣≥2. Note that the dual graph G0 of D consists of two vertices and a single edge joining them. Let G be the graph obtained by adding 2∣f1∣−1 parallels of the edge, that is, G consists of 2∣f1∣ edges between the two vertices. Apply the same construction as above, using this G, to triangulate M. In this case, we have 2∣f1∣ type (ii) pieces and no type (i) pieces. Using ∣f1∣≥2, it is verified that our construction produces a simplicial complex structure. (No two vertices of a tetrahedron are identified and each tetrahedron is uniquely determined by its vertices.) By the above counting, the number of tetrahedra is 48∣f1∣, as desired.□ Figure 3. View largeDownload slide Prism decomposition of Δ1×Δ2. Figure 3. View largeDownload slide Prism decomposition of Δ1×Δ2. Figure 4. View largeDownload slide A decomposition of a link diagram. Figure 4. View largeDownload slide A decomposition of a link diagram. Figure 5. View largeDownload slide Decomposition of subpieces. Figure 5. View largeDownload slide Decomposition of subpieces. Figure 6. View largeDownload slide A boundary component and a Dehn filling torus. Figure 6. View largeDownload slide A boundary component and a Dehn filling torus. 3. Linear complexity triangulations from Heegaard splittings In this section, we present an explicit construction of a triangulation from a Heegaard splitting given by a mapping class. Recall from Definition 1.2 that the Heegaard–Lickorish complexity of a closed 3-manifold M is the minimal word length, in the Lickorish generators, of a mapping class on an arbitrary surface which gives a Heegaard splitting of M. Here the Lickorish generators of the mapping class group Mod(Σg) of an oriented surface Σg of genus g are defined to be the ±1 Dehn twists along the curves α1,…,αg, β1,…,βg, γ1,…,γg−1 as shown in Fig. 1. To make it precise, we use the following convention. Fix a standard embedding of a surface Σg of genus g in S3 as in Fig. 1. Then Σg bounds the inner handlebody H1 and the outer handlebody H2 in S3. Let ij:Σg→Hj (j = 1,2) be the inclusion. The mapping class h∈Mod(Σg) of a homeomorphism f:Σg→Σg gives a Heegaard splitting (Σg,{βi},{f(αi)}) of the 3-manifold M=(H1∪H2)/i1(f(x))∼i2(x),x∈Σg. In other words, M is obtained by attaching g2-handles to the inner handlebody H1 with boundary Σg along the curves f(αi) and then attaching a 3-handle. Under our convention, the identity mapping class gives us S3. The Heegaard–Lickorish complexity can be compared with the Heegaard genus by the following lemma. Lemma 3.1 Suppose M is a closed 3-manifold with a Heegaard splitting given by a mapping class h∈Mod(Σg),which is a product of ℓ Lickorish generators. Then for some g′≤2ℓ, M admits a Heegaard splitting given by a mapping class h′∈Mod(Σg′),which is a product of ℓ Lickorish generators. From Lemma 3.1, it follows immediately that the Heegaard genus is not greater than twice the Heegaard–Lickorish complexity. On the other hand, it is easily seen that a 3-manifold may be drastically more complicated than another with the same Heegaard genus. For example, all the lens spaces L(n,1) have Heegaard genus one, but L(n,1) is represented by a genus one mapping class of Heegaard–Lickorish word length n. In fact, by results of [4] (see also Lemma 4.2 and related discussions in the present paper), csimp(L(n,1))→∞ as n→∞, and consequently cHL(L(n,1))→∞ and csurg(L(n,1))→∞ by Theorems A and B. Proof of Lemma 3.1 For a Lickorish generator t∈Mod(Σg), we say that t passes through the ith hole of Σg if t is a Dehn twist along either one of the curves αi, βi, γi or γi−1 (see Fig. 1). It is easily seen from Fig. 1 that a Lickorish generator can pass through at most two holes of Σg. Therefore, the Lickorish generators which appear in the given word expression of h of length ℓ can pass through at most 2ℓ holes. If g>2ℓ, then for some i, no Lickorish generator used in h passes through the ith hole. By a destabilization which removes the ith hole from Σg, we obtain a Heegaard splitting of M of genus g−1 given by a mapping class, which is a product of ℓ Lickorish generators. By an induction, the proof is completed.□ Lickorish’s work [10, 11] presents a construction of a surgery presentation from a Heegaard splitting. From his proof, we obtain the following: Theorem 3.2 For any closed 3-manifold M, csurg(M)≤2·cHL(M)2+3·cHL(M). Proof Suppose M has a Heegaard splitting represented by a mapping class of Lickorish word length ℓ. By the arguments in Lickorish [10, 11] (see also Rolfsen’s book [17, Chapter 9, Section I]), M is obtained by surgery on a link L with ℓ(±1)-framed components, which admits a planar diagram in which no component has a self-crossing, and any two distinct components have at most two crossings between them. See Fig. 7 for an example. It follows that n(L)=0, f(L)=ℓ and c(L)≤2·(ℓ2)=ℓ(ℓ+1). By definition, we have csurg(M)≤2c(L)+f(L)+n(L)≤2ℓ2+3ℓ.□ Remark 3.3 Conversely, a surgery presentation can be converted to a Heegaard splitting. For instance, Lu’s method in [12] tells us how to obtain a Heegaard splitting from a surgery link, as a product of explicit Dehn twists on an explicit surface. By rewriting those Dehn twists in terms of the Lickorish twists, for instance by following the arguments of existing proofs that Lickorish twists generate the mapping class group (for example, see [10, 11] or [6]), one would obtain a word in the Lickorish twists which represents the mapping class, and in turn an upper bound for the Heegaard–Lickorish complexity of the 3-manifold. We do not address details here. Remark 3.4 Theorem 3.2 and (the proof of) Theorem B immediately give a triangulation from a Heegaard splitting, together with the following complexity estimate: csimp(M)≤72·(2·cHL(M)2+3·cHL(M)). It tells us that the simplicial complexity is bounded by a quadratic function in the Heegaard–Lickorish complexity. A quadratic bound seems to be the best possible result from this method (unless one finds a clever simplification of the resulting surgery link). For instance, by generalizing the rightmost five components in Fig. 7 and considering the corresponding mapping class, one sees that there is actually a genus one mapping class of Lickorish word length ≤ℓ for which the associated link L has crossing number ≥ℓ2(ℓ2−1). In general, except for sufficiently small values of cHL, this quadratic bound is weaker than the linear bound in Theorem A. Remark 3.5 The upper bound to the (pseudo-)simplicial complexity in terms of Heegaard splittings given in Theorem A is often stronger than Matveev’s upper bound in [13, 15]. We recall Matveev’s result: suppose M admits a Heegaard splitting M=H1∪ΣH2 with handlebodies H1 and H2 and Heegaard surface Σ. Let α and β be the union of the meridian curves of H1 and H2 on Σ, respectively. Suppose α and β are transverse, n=#(α∩β), and the closure of a component of Σ⧹(α∪β) contains m points in α∩β. Then, c(M)≤n−m [13, Proposition 3; 15, Proposition 2.1.8]. As an explicit example, let τ, σ be the +1 Dehn twists along the meridian and preferred longitude on the boundary of the standard solid torus in S3, and consider the lens space L with Heegaard splitting determined by the mapping class of σkτk. It is straightforward to see that n=k2+1 and m = 4 for this Heegaard splitting, so that the result in [13, 15] gives c(L)≤k2−3, a quadratic upper bound. On the other hand, Theorem A gives a linear upper bound c(L)≤csimp(L)≤1104k, since σkτk has Lickorish word length ≤2k. In fact, for arbitrary N>0, we can construct examples of lens spaces, using mapping classes of the form (σkτk)N and τk(σkτk)N, for which Matveev’s upper bound c(M)≤n−m has order N (that is, asymptotic growth of kN), while Theorem A gives a linear upper bound. Figure 7. View largeDownload slide An example of Lickorish’s surgery link. Figure 7. View largeDownload slide An example of Lickorish’s surgery link. The rest of this section is devoted to the proof of Theorem A. The key idea used in our proof below, which enables us to produce a more efficient triangulation (cf. Remark 3.4), is that we view Lickorish’s surgery link (Fig. 7) as a link in the thickened Heegaard surface. Proof of Theorem A Here we will prove the following statement, which is slightly sharper than Theorem A: if a closed 3-manifold M≠S3 has Heegaard–Lickorish complexity ℓ, then the simplicial complexity of M is not greater than 552ℓ−120. Suppose h∈Mod(Σg) gives a Heegaard splitting of a given 3-manifold M, and suppose h is a product of ℓ Lickorish generators. Both g and ℓ are non-zero, since M≠S3. By Lickorish [10], M is obtained by surgery on an ℓ-component link L in S3, where each component has either (+1) or (−1)-framing. His proof tells us more about L (another useful reference for this is [17, Chapter 9, Section I]). In fact, L lies in a bicollar Σg×[−1,1] of Σg in S3, and each component is of the form αi×{t}, βi×{t}, or γi×{t} for some i and t∈[−1,1]. An example is shown in Fig. 7. Let D=(⋃i=1gαi)∪(⋃i=1gβi)∪(⋃i=1g−1γi). Then, L lies on D×[−1,1]⊂S3. Note that for a link in the bicollar Σg×[−1,1], if each component is regular with respect to the projection of Σg×[−1,1]→Σg, then the blackboard framing with respect to Σg is well defined; the preferred parallel with respect to the blackboard framing is defined to be the push-off along the [−1,1] direction. In particular, for our surgery link L, the blackboard framing with respect to Σg is equal to the zero framing in S3. Now, in order to construct a triangulation of Σg×[−1,1]⧹ν(L), we proceed similarly to the proof of Lemma 2.1; the difference is that we now use a ‘diagram’ on Σg, instead of a planar link diagram. Let G0 be the dual graph of D on Σg. Let G be the graph shown in Fig. 8, which is obtained by adding parallel edges to G0. Note that for each of the curves αi, βi and γi, an edge of G0 dual to the curve is chosen and two parallels of the chosen edge are added to produce G. Each region of G is a quadrangle or a bigon. (Each quadrangle/bigon has no two edges which are identified, while vertices are allowed to be identified; using this, it can be verified that our construction described below gives a simplicial complex structure in which each tetrahedron has no identified vertices and is uniquely determined by its vertices.) Cutting Σg×[−1,1]⧹ν(L) along G×[−1,1], we obtain pieces corresponding to quadrangle regions and bigon regions; call them type (i) and (ii), respectively. See the left of Fig. 9. Cutting along D×[−1,1], a type (i) piece is divided into four cubic subpieces, and a type (ii) piece is divided into two triangular prism subpieces. See the middle of Fig. 9. Hatched quadrangles represent ∂ν(L). For a type (i) subpiece, triangulate the three front faces of each subpiece as in the top right of Fig. 9, and then triangulate the subpiece by taking a cone at the opposite vertex, as we did in the proof of Lemma 2.1. We claim that there are 6k+6 tetrahedra in this subpiece triangulation, where k is the number of hatched quadrangles in the subpiece. The number of tetrahedra in the subpiece is equal to the number of triangles in the three front faces. There are two triangles in the top face. To count triangles in the remaining two faces, observe that the front middle vertical edge is divided into 2k+1 1-simplices. There are 4k+2 triangles that have one of these 1-simplices as an edge, and there are 2k+2 remaining triangles. Therefore, there are total 6k+6 triangles, as we claimed. A type (ii) subpiece is triangulated similarly, as depicted in the bottom of Fig. 9. When a type (ii) subpiece has k hatched quadrangles, its triangulation has 4k+3 tetrahedra. Combining the triangulations of the subpieces, we obtain a triangulation of Σg×[−1,1]⧹ν(L). To estimate the number of tetrahedra, first observe that the graph D has 3g−2 vertices, where g is the genus of the Heegaard surface Σg. Therefore, its dual graph G0 has 3g−2 quadrangular regions. Since 2(3g−1) parallel edges have been added to G0 and each of them introduces a bigon region, the graph G has 3g−2 quadrangular regions and 6g−2 bigon regions. It follows that there are 12g−8 type (i) subpieces and 12g−4 type (ii) subpieces in Σg×[−1,1]⧹ν(L). Also, observe that each component of L passes through type (i) pieces at most three times, and type (ii) pieces twice. Therefore, a component can contribute at most 4·3=12 hatched quadrangles in type (i) subpieces, and 2·2=4 hatched quadrangles in type (ii) subpieces. It follows that there are at most 6·12ℓ+6·(12g−8)+4·4ℓ+3·(12g−4)=88ℓ+108g−60 tetrahedra in our triangulation of Σg×[−1,1]⧹ν(L). For later use, note that our triangulation restricted to Σg×{t} ( t=±1) has 2(12g−8)+(12g−4)=36g−20 triangles, since the top face of each of the 12g−8 type (i) subpieces consists of two triangles, and the top of each of the 12g−4 type (ii) subpieces is a single triangle. Now we triangulate the inner and outer handlebodies, which are the components of S3⧹(Σg×(−1,1)). First we consider the outer handlebody. Choose disjoint disks D0,D1,…,Dg in the outer handlebody such that ∂Di=αi for i = 1, …,g, and ∂D0 is the union of the outermost edges of the graph G in the top view of Fig. 8; ∂D0 is parallel to the outer dotted circle in Fig. 8. Our triangulation on Σg×{1} divides ∂D0 into 2g edges, each of ∂D1 and ∂Dg into six edges, and each ∂Di ( i=2,…,g−1) into eight edges. Extending this triangulation of the boundary, we triangulate D0 into 2g−2 triangles, each of D1, Dg into four triangles, and each Di ( i=2,…,g−1) into six triangles, by drawing edges joining vertices. Cutting the outer handlebody along the disks D0, …, Dg, we obtain two 3-balls B1 and B2. Our triangulations of the Di and Σg×{1} give triangulations of ∂B1 and ∂B2. Triangulate each of B1 and B2 by taking the cone of the boundary. Note that a triangle in Σ×{1} contributes one tetrahedron to B1∪B2, while a triangle in Di contributes two tetrahedra to B1∪B2. It follows that the outer handlebody has at most (36g−20)+2·(2g−2+6g−4)=52g−32 tetrahedra. For the inner handlebody, choose disjoint disks D1′,…,Dg′,D1″,…,Dg−1″ in the inner handlebody such that ∂Di′=βi and ∂Di″=γi. Similar to the case of the disks Di above, our triangulation extends to (⋃Di′)∪(⋃Dj″), where Di′ and Di″ are decomposed to two and four triangles, respectively. Cutting the inner handlebody along the disks Di′ and Di″, we obtain g3-balls. Triangulate each 3-ball by taking the cone of the boundary. A counting argument similar to the above shows that the inner handlebody has (36g−20)+2(2g+4(g−1))=48g−28 tetrahedra. To obtain the surgery manifold, attach and triangulate Dehn filling tori as in the proof of Lemma 2.1. Recall that the blackboard framing is equal to the zero framing in the present case. Since each component of L passes through two type (ii) pieces, each of which introduces a half twist with respect to the blackboard framing, each Dehn filling torus can be assumed to be attached along the given (±1)-framing of L, by appropriately choosing diagonal edges used to triangulate the hatched quadrangles of type (ii) pieces in Fig. 9. Therefore, the surgery manifold is equal to the given M. Since there are at most 16ℓ hatched quadrangles, and each hatched quadrangle contributes a triangular prism which consists of 3 tetrahedra in the Dehn filling tori, there are at most 48ℓ tetrahedra in the Dehn filling tori. It follows that our triangulation of the surgery manifold M has at most (88ℓ+108g−60)+(52g−32)+(48g−28)+48ℓ=136ℓ+208g−120 tetrahedra. By Lemma 3.1, we may assume that g≤2ℓ. It follows that the simplicial complexity of M is at most 552ℓ−120.□ Figure 8. View largeDownload slide The graphs D and G on Σg, which are depicted in thick and thin edges, respectively. Figure 8. View largeDownload slide The graphs D and G on Σg, which are depicted in thick and thin edges, respectively. Figure 9. View largeDownload slide Decomposition of the surgery link exterior pieces. Figure 9. View largeDownload slide Decomposition of the surgery link exterior pieces. 4. Theorems A and B are asymptotically optimal In this section, we prove Theorem D and related results. For this purpose, we use some results in [4]. First, we need the following lower bound of the simplicial complexity. In [2], Cheeger and Gromov introduced the von Neumann L2ρ-invariant ρ(2)(M,ϕ)∈R which is defined for a smooth closed (4k−1)-manifold M and a homomorphism ϕ:π1(M)→G. By deep analytic arguments, they showed that for each M, there is a universal bound for the values of ρ(M,ϕ) [2]; that is, there is CM>0 satisfying that ∣ρ(2)(M,ϕ)∣≤CM for any ϕ. In [4], a topological approach to the universal bound for ρ(2)(M,ϕ) is presented, and in particular, an explicit linear universal bound is given in terms of the simplicial complexity of 3-manifolds: Theorem 4.1 [4, Theorem 1.5] Suppose M is a closed 3-manifold. Then ∣ρ(2)(M,ϕ)∣≤363 090·csimp(M)for any homomorphism ϕ. In this paper, we will use the Cheeger–Gromov ρ-invariant as a lower bound of the simplicial complexity. For the lens space L(n,1) and the identity map id:π1(L(n,1))→Zn ( n>0), [4, Lemma 7.1] gives the following value of the Cheeger–Gromov ρ-invariant, using the computation of Atiyah–Patodi–Singer [1, p. 412]: ρ(2)(L(n,1)),id=n3+23n−1. From this and Theorem 4.1, a lower bound of the simplicial complexity of L(n,1) is obtained. We state it as a lemma: Lemma 4.2 csimp(L(n,1))≥n−31 089 270. We remark that a pseudo-simplicial complexity analogue is given in [4, Corollary 1.15]. Now we are ready to proof Theorem D. In fact, the following stronger inequalities hold, and Theorem D follows immediately from them. Theorem 4.3 11 089 720·(1−3n)·cHL(L(n,1))≤csimp(L(n,1)),11 089 720·(1−3n)·csurg(L(n,1))≤csimp(L(n,1)). Proof Since L(n,1) is obtained by the n-framed surgery on the unknot, it is easily seen that cHL(M), csurg(M)≤n. The desired inequalities follow from this and Lemma 4.2.□ In what follows, we discuss a generalization and a specialization of the lens space case we considered in Theorem 4.3. First, the second inequality in Theorem 4.3 generalizes for a larger class of 3-manifolds. For a knot K in S3, let M(K,n) be the 3-manifold obtained by n-framed surgery on K. Let g4(K) be the (topological) slice genus of K. Theorem 4.4 For any n≠0, 11 089 720·(1−3+6g4(K)∣n∣)·(csurg(M(K,n))−2c(K))≤csimp(M(K,n)). Proof Let ϕ:π1(M(K,n))→Z∣n∣ be the abelianization. Due to [3, Equation (2.8)], ∣ρ(2)(M(K,n),ϕ)∣≥13·(∣n∣−3−6g4(K)). By Theorem 4.1, it follows that csimp(M(K,n))≥11 089 270·(∣n∣−3−6g4(K)). (4.1) By definition, csurg(M(K,n))≤2c(K)+∣n∣. From this and (4.1), the desired inequality follows.□ On the other hand, if we consider the special case of lens spaces L(2k,1), then the inequalities in Theorem 4.3 (and hence those in Theorem D) can be improved significantly as follows. Theorem 4.5 For k>1, the following hold: (1−32k)·cHL(L(2k,1))≤csimp(L(2k,1)),(1−32k)·csurg(L(2k,1))≤csimp(L(2k,1)). Proof Due to Jaco et al. [7], the pseudo-simplicial complexity of L(2k,1) is equal to 2k−3 for k>1, and consequently csimp(L(2k,1))≥2k−3. Using this in place of Lemma 4.2 in the proof of Theorem 4.3, we obtain the inequalities.□ We finish this section with a proof of Theorem C. Proof of Theorem C Recall the definition of the ‘largest possible value’ of the simplicial complexity for Heegaard–Lickorish complexity ≤ℓ: sHL(ℓ)≔sup{csimp(M)∣cHL(M)≤ℓ}. The first assertion of Theorem C, which says sHL(ℓ)∈O(ℓ)∩Ω(ℓ), follows immediately from the estimate 11 089 270≤limsupℓ→∞sHL(ℓ)ℓ≤552, (4.2) which we prove in what follows. Fix ℓ. For any M with cHL(M)≤ℓ, we have csimp(M)ℓ≤csimp(M)cHL(M)≤552 by Theorem A. Taking the supremum over all such M, we obtain sHL(ℓ)/ℓ≤552. From this, we obtain the upper bound in (4.2). By the definition of sHL(ℓ), we have csimp(M)cHL(M)≤sHL(cHL(M))cHL(M) for any M. By Theorem 4.3, the limit supremum of the left-hand side as cHL(M)→∞ is bounded from below by 1/1 089 270. From this, the lower bound in (4.2) follows. The analogous statement for the function ssurg(k) is proved by the same argument.□ Acknowledgements The author thanks an anonymous referee for comments which were very helpful in improving results and in fixing a mistake of an earlier version of this paper. Funding This work was partially supported by NRF Grants 2013067043 and 2013053914. References 1 M. F. Atiyah , V. K. Patodi and I. M. Singer , Spectral asymmetry and Riemannian geometry. II , Math. Proc. Cambridge Philos. Soc. 78 ( 1975 ), 405 – 432 . Google Scholar CrossRef Search ADS 2 J. Cheeger and M. Gromov , Bounds on the von Neumann dimension of L2-cohomology and the Gauss–Bonnet theorem for open manifolds , J. Diff. Geom. 21 ( 1985 ), 1 – 34 . Google Scholar CrossRef Search ADS 3 J. C. Cha , Complexity of surgery manifolds and Cheeger–Gromov invariants , Int. Math. Res. Not. IMRN 2016 ( 2016 ), 5603 – 5615 . Google Scholar CrossRef Search ADS 4 J. C. Cha , A topological approach to Cheeger–Gromov universal bounds for von Neumann ρ-invariants , Commun. Pure Appl. Math. 69 ( 2016 ), 1154 – 1209 . Google Scholar CrossRef Search ADS 5 J. C. Cha and C. Livingston , KnotInfo: table of knot invariants, http://www.indiana.edu/~knotinfo. 6 B. Farb and D. Margalit , A primer on mapping class groups, Princeton Mathematical Series vol. 49 , Princeton University Press , Princeton, NJ , 2012 . 7 W. Jaco , H. Rubinstein and S. Tillmann , Minimal triangulations for an infinite family of lens spaces , J. Topol. 2 ( 2009 ), 157 – 180 . Google Scholar CrossRef Search ADS 8 W. Jaco , J. H. Rubinstein and S. Tillmann , Coverings and minimal triangulations of 3-manifolds , Algebr. Geom. Topol. 11 ( 2011 ), 1257 – 1265 . Google Scholar CrossRef Search ADS 9 W. Jaco , J. H. Rubinstein and S. Tillmann , Z2-Thurston norm and complexity of 3-manifolds , Math. Ann. 356 ( 2013 ), 1 – 22 . Google Scholar CrossRef Search ADS 10 W. B. R. Lickorish , A representation of orientable combinatorial 3-manifolds , Ann. Math. 76 ( 1962 ), 531 – 540 . Google Scholar CrossRef Search ADS 11 W. B. R. Lickorish , A finite set of generators for the homeotopy group of a 2-manifold , Proc. Cambridge Philos. Soc. 60 ( 1964 ), 769 – 778 . Google Scholar CrossRef Search ADS 12 N. Lu , A simple proof of the fundamental theorem of Kirby calculus on links , Trans. Amer. Math. Soc. 331 ( 1992 ), 143 – 156 . Google Scholar CrossRef Search ADS 13 S. V. Matveev , Complexity theory of three-dimensional manifolds , Acta. Appl. Math. 19 ( 1990 ), 101 – 130 . 14 S. V. Matveev , Complexity of three-dimensional manifolds: problems and results [translated from proceedings of the conference ‘geometry and applications’ dedicated to the seventieth birthday of V. A. toponogov (russian) (novosibirsk, 2000), 102–110, Ross. Akad. Nauk Sib. Otd. Inst. Mat., Novosibirsk, 2001], Siberian Adv. Math. 13 ( 2003 ), 95–103. 15 S. Matveev , Algorithmic topology and classification of 3-manifolds , Algorithms and Computation in Mathematics, vol. 9, 2nd edn , Springer , Berlin , 2007 . 16 S. Matveev , C. Petronio and A. Vesnin , Two-sided asymptotic bounds for the complexity of some closed hyperbolic three-manifolds , J. Aust. Math. Soc. 86 ( 2009 ), 205 – 219 . Google Scholar CrossRef Search ADS 17 D. Rolfsen , Knots and links , Mathematics Lecture Series, No. 7. Publish or Perish Inc. , Berkeley, CA , 1976 . 18 J. Weeks , Computation of hyperbolic structures in knot theory, Handbook of Knot Theory , Elsevier B. V , Amsterdam , 2005 , 461 – 480 . Google Scholar CrossRef Search ADS © 2017. Published by Oxford University Press. All rights reserved. For permissions, please email: journals.permissions@oup.com This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/about_us/legal/notices)

The Quarterly Journal of Mathematics – Oxford University Press

**Published: ** Oct 10, 2017

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