arXiv:1509.07647v4 [math.GT] 15 Oct 2019 h eta Universities. Central the h oki upre yteFnaetlRsac ud lzujbk Funds Research Fundamental the by supported is work The phrases. and words Key 2010 Date .Pofo h anterm12 13 cover open of nerve and References Triangulation balls of theorem 7.2. Volume main the of 7.1. Proof nerve and cover 7. Good 6. .Cmlxt f3mnfls6 4 7 and complexes inequality Cubical isoperimetric and embedding 5. Kuratowski 3-manifolds of 4. manifolds Complexity aspherical of volume 3. Systolic 2. Introduction 1. YTLCVLM N OPEIYOF COMPLEXITY AND VOLUME SYSTOLIC coe 6 2019. 16, October : ahmtc ujc Classification. Subject Mathematics mmnme fttaer natinuain hc sanatural a complicatedness. is combinatorial which the triangulation, evaluate a min- to in the tool is tetrahedra 3-manifolds of 3-manifolds of number complexity of imum The volume complexity. systolic to that work related show is we to paper, spirit this Gromov’s In topolog- along some complicatedness. to precisely, measuring related sys- invariants More is ical volume the manifold. systolic the the that that of proved is proved topology Gromov volume a on Gromov depends in systolic Moreover, volume constant the tolic invariant. optimal that homotopy the showed com- as a Babenko of defined terms inequality. is in systolic volume below Systolic bounded is plexity. 3-manifold aspherical closed a Abstract. nti ae,w rv httessoi oueof volume systolic the that prove we paper, this In ytlcvlm,cmlxt,apeia 3-manifolds. aspherical complexity, volume, Systolic 3-MANIFOLDS IH CHEN LIZHI Contents δ etnin9 -extension 1 rmr 32,Scnay57M27. Secondary 53C23, Primary -072 for y-2017-26 17 12 10 2 2 L. CHEN
1. Introduction Let M be a closed aspherical manifold endowed with a Riemann- ian metric , denoted (M, ). The homotopy 1-systole of (M, ), de- G G G noted Sys π1(M, ), is the shortest length of a noncontractible loop in M. Gromov [7, TheoremG 0.1.A.] proved that on a closed asphercial n- manifold M, every Riemannian metric satisfies the following systolic inequality, G n Sys π1(M, ) 6 Cn Vol (M), (1.1) G G where Cn is a constant only depending on the manifold dimension n. Gromov [9, Section 4.26.] further verified that the constant Cn is re- lated to the topology of the manifold M, n Sys π1(M, ) 6 Top Vol (M), (1.2) G | | G where Top is some measure of the topological complexity of M. In this paper,| | we work along Gromov’s spirit to show that the constant Top is dependent on complexity defined in terms of triangulation of a| closed| aspherical 3-manifold. Examples of systolic inequalities of type (1.2) include the case of closed hyperbolic surfaces Σg of genus g, 2 2 log g Sys π1(Σg, ) 6 C Area (Σ), G g G where C is a fixed constant. The constant Top here depends on the genus g. A generalization to higher dimensional| aspherical| n-manifolds M is realized by using simplicial volume, 3 n log (1 + M ) Sys π1(M, ) 6 Cn k k Vol (M), (1.3) G M G k k where Cn is a constant only depending on the dimension n. The simpli- cial volume M is defined as Gromov norm of the fundamental class [M], that is,k k M = inf a , k k | i| Xi where the infimum is taken over all cycles i aiσi representing the R fundamental class [M] in Hn(M; ). Moreover,P the constant Top in (1.2) can be dependent on simplicial height h(M), | |
exp Cn′ log h(M) n Sys π1(M, ) 6 Cn p Vol (M), (1.4) G h(M) G where Cn and Cn′ are constants only depending on the dimension n, see Gromov [8, Section 3.C.3.] and [9, Section 4.26.]. The simplicial height SYSTOLIC VOLUME AND COMPLEXITY OF 3-MANIFOLDS 3 h(M) of a closed manifold M is defined to be the minimum number of simplices in a geometric cycle representing the fundamental class [M]. Moreover, Gromov [8, Section 3.C.3.] also defined modified simplicial height h+(M) of a closed aspherical manifold M as inf λ , | i| Xi where the infimum is taken over all integral cycles i aiσi representing the fundamental class [M]. He further mentioned inP [8, Section 3.C.3.] that it is desirable to have an inequality of type similar to (1.3) or (1.4). In this paper, we provide a partial answer to this problem in dimension three. Matveev [21] defined complexity of closed 3-manifolds. The complexity is a topological invariant equal to the minimum number of tetrahedra in a triangulation, if M is a closed irreducible 3-manifold other than S3, RP3 and the lens space L(3, 1), see Section 3. Hence according to the above definitions, Gromov’s modified simplicial height coincides with complexity, if the closed 3-manifold M is irreducible and not S3, RP3 and L(3, 1). In this paper, we show that the modified simplicial height, or complexity, satisfies inequality of the form like (1.4), if M is a 3-manifold described above. The optimal constant in a systolic inequality is usually called systolic volume, see [24, 25]. Definition 1.1. The systolic volume of a closed aspherical n-dimensional manifold M, denoted by SR(M), is defined as Vol (M) inf G , Sys π (M, )n G 1 G where the infimum is taken over all Riemannian metrics on M. G It is proved by Babenko [3] that the systolic volume SR(M) is a homotopy invariant. The main theorem of this paper is that the systolic volume of a closed aspherical 3-manifold has a lower bound depending on complexity. Theorem 1.2. Let M be a closed aspherical 3-manifold. The systolic volume SR(M) satisfies c(M) SR(M) > C , (1.5) exp C′ log c(M) p where C and C′ are two constants which can be explicitly calculated. In the proof of Theorem 1.2, we construct nerve of open cover of a smooth triangulation K of M. The covering argument and method of 4 L. CHEN nerves are used in Gromov [7] ( also see Guth [12] ). In terms of the construction of nerve, we can have a lower bound of systolic volume depending on the number of simplices in a polyhedron. The Alexan- droff map from the triangulation K to the nerve can be made to be simplicial, so that there is a comparison of the simplices in triangula- tion K and the nerve. Hence we get the estimate of systolic volume in terms of complexity. The balls B(x, r) K in the cover used to construct nerve has the following property, ⊂ Vol(B(x, r)) > Crn 1 holds when r (0, 2 Sys π1(K)), where B(x, r) is the ball with center x and radius r∈, and C is a constant. In order to have such an estimate of volume of balls, we embed K isometrically into the Banach space L∞. Here L∞ is the space of all bounded Borel functions on K, which appears in Gromov’s proof of the systolic inequality, see [7]. Such an embedding is called Kuratowski embedding. Moreover, there is a finite approximation of the embedding. The estimate of volume of balls holds since there are isoperimetric inequalities in the Banach space L∞, see [7] and [1]. This paper is organized as follows. Section 2 contains a brief review to the present research progress of the systolic volume. In Section 3, we introduce the complexity invariant of 3-manifolds. In Section 4, Gromov’s work on Kuratowski embedding and isoperimetric inequality in Banach spaces are introduced. They are used in the proof of the main theorem of this paper. In Section 5, we explain how to construct cubic complexes and δ-extensions in the Banach space. In Section 6, good open cover and the associated nerve is presented. Construction of the nerve is a main technique in the proof of main theorem. The proof of main Theorem 1.2 is given in Section 7.
2. Systolic volume of aspherical manifolds Study of systolic volume occupies a main part in the research of systolic geometry, see [6] or [18]. Problems related to the systolic vol- ume include calculating exact values, optimal metrics realizing systolic volume, topological properties related to systolic volume. Note that in literature, the systolic volume is sometimes called optimal systolic ratio ( see Croke and Katz [6]), or systolic constant ( see Brunnbauer [5], Elmir and Lafontaine [22]). Exact values of systolic volume of surfaces is known for 2-torus T2 √3 RP2 2 SR( ) = 2 , real projective plane SR( ) = π and Klein bottle RP2 RP2 2√2 SR( # ) = π . Moreover, among all closed manifolds with SYSTOLIC VOLUME AND COMPLEXITY OF 3-MANIFOLDS 5 nonzero systolic volume, we currently only know exact values of sys- tolic volume for these three examples. In 2-dimension, a closed non-simply connected surface Σ satisfies 2 SR(Σ) > , π where equality holds if Σ is a real projective plane RP2, see Croke and Katz [6]. Morevoer, if Σg is a closed orientable hyperbolic surface of genus g, g 9π g π . SR(Σg) . (2.1) (log g)2 4 (log g)2 holds when the genus g is large enough, see [19]. More estimates of systolic volume for surfaces can be found in Croke and Katz [6, Figure 2.1.]. In higher dimensions, Gromov [7] proved that 1 SR(M) > n 6(n + 1)nn (n + 1)! p holds for aspherical n-dimensional manifolds M. Wenger’s work [26] made an improvement, 1 SR(M) > . (6 27nn!)n · Guth [11] further improved the lower bound estimate of systolic volume, 1 SR(M) > , (8n)n where M is n-torus Tn, or real projective n-space RPn, or n-manifolds with nice cohomology properties. For the relation between systolic volume and other topological in- variants, Gromov also showed Top in (1.2) is dependent on Betti numbers b(M), | | b(M) SR(M) > Cn , exp Cn′ log b(M) p where Cn and Cn′ are constants only depending on the dimension n. Sabourau extend this result to the connected sum of aspherical mani- folds, k SR(#kM) > Cn , (2.2) exp (Cn′ √log k) where #kM stands for the connected sum of k copies of the manifold M, and Cn and Cn′ are two constants only depending on the dimension 6 L. CHEN n of the manifold M. Babenko and Balacheff [4] proved an upper bound to the connected sum, k log (log k) SR(# M) 6 C , k p√log k with C a constant only depending on the manifold M. In addition, they conjectured that the systolic volume satisfies k SR(# M) > C , k logn k where #kM is the connected sum of k copies of the aspherical n- manifold M, with n > 3, see [4]. Sabourau [24] and Brunnbauer [5] showed that the systolic volume of aspherical n-manifolds has lower bound in terms of minimal entropy λ(M), λ(M) SR(M) > C , n logn (1 + λ(M)) which is another example to the type of inequality (1.2).
3. Complexity of 3-manifolds The complexity of 3-manifolds can be considered as a natural topo- logical invariant to evaluate how complicated the manifold is. Matveev [21] defined complexity of a compact 3-manifold M as the minimal number of vertices in a special spine of M. For a closed irre- ducible 3-manifold other than S3, RP3 and the lens space L(3, 1), the complexity is equal to the minimum number of tetrahedra in a trian- gulation. In this paper, we define the complexity of closed aspherical 3-manifolds to be minimum number of tetrahedra in a triangulation. A triangulation of a 3-manifold M, denoted ( , h), is a simplicial complex with a homeomorphism h : M,T where stands for the unionT of all simplices of . The union|T|→ is called the|T | underlying polyhedron of the triangulation.T |T | Definition 3.1 (Pseudo-simplicial triangulation, see [14]). A pseudo- simplicial triangulation of a 3-manifold M contains T (1) a set ∆= ∆˜ i of disjoint collection of tetrahedra, (2) a family Φ{ of} isomorphisms pairing faces of the tetrahedra in ∆ so that if φ Φ, then φ is an orientation-reversing affine ∈ isomorphism from a faceσ ˜i ∆˜ i to a faceσ ˜j ∆˜ j, possibly i = j. ∈ ∈ SYSTOLIC VOLUME AND COMPLEXITY OF 3-MANIFOLDS 7
Denote by ∆/Φ the space obtained from the disjoint union of the ∆˜ i by setting x σ˜i equal to φ(x) σ˜j, with the identification topology. The manifold∈M is homeomorphic∈ to = ∆/Φ. |T | In a pseudo-simplicial triangulation, two faces of the same tetrahedra possibly are identified. Definition 3.2. Let M be a closed irreducible 3-manifold. The com- plexity of M, denoted c(M), is the minimal number of tetrahedra in a pseudo-simplicial triangulation. The complexity c(M) has the following finiteness property and ad- ditivity property. Proposition 3.3 (Matveev [21, Theorem A, B]). (1) For any in- teger k, there exist only a finite number of distinct closed ir- reducible orientable 3-manifolds of complexity k. (2) The complexity of the connected sum of compact 3-manifolds is equal to the sum of their complexities.
The complexity of a 3-manifold measures how complicated a combi- natorial description of the manifold must be, see [21]. It is difficult to determine the complexity of 3-manifolds. There are various investiga- tions of upper bounds estimate and lower bounds estimate to complex- ity. Moreover, exact values of complexity are calculated for some family of infinitely many 3-manifolds. For example, Matveev et al. [23] proved two-sided bounds for some families of hyperbolic 3-manifolds. Anisov [2] calculated exact values of complexity for a family of infinitely many 3-manifolds with Sol geometry. More results are included in a series of papers by Jaco, Rubinstein and Tillmann, see [15, 16, 17]. Result of this paper can be seen as one more way to the study of complexity.
4. Kuratowski embedding and isoperimetric inequality A manifold with a distance function can be embedded into a metric space by Kuratowski embedding. Gromov [7] proved the systolic in- equality (1.1) by using Kuratowski embedding and generalized isoperi- metric inequality in Banach spaces. Let M be a closed connected n-dimensional manifold. The set of all bounded Borel functions f on M, denoted L∞(M), is a Banach space with the norm
f = sup f(x) , f L∞(M). k k∞ x M | | ∈ ∈ 8 L. CHEN
Given a distance function distM , the manifold M becomes a metric space. Define the map K : M L∞(M) by setting →
K(x)(w)= distM (x, w) for any w M. ∈ Lemma 4.1 (Gromov 1983, [7] Section 2.). For any v,w M, the following equality holds, ∈
dist(v,w)= K(v) K(w) . (4.1) k − k∞
Lemma 4.1 implies that K : M L∞(M) is an embedding. → Definition 4.2. The map K : M L∞(M) is called Kuratowski embedding. →
The Kuratowski embedding K : M L∞(M) can be approximated → by a finite dimensional embedding. Let M0 M be an ε-net. Then there exists a (1 + ε) bi-Lipschitz finite dimensional⊂ embedding of M, see [1, Proposition 5.1.]. Gromov defined filling volume of a manifold in [7]. Let [M] be the fundamental class of M. For a closed manifold M with a distance function defined on it, the filling volume of a submanifold V M, denoted FillVol(V M), is defined to be the lower bound of volumes⊂ of the cycles z with⊂∂z = [V ]. Definition 4.3. Let M be a closed manifold endowed with a distance function. The filling volume of M, denoted Fill Vol(M), is defined as
FillVol(M L∞(M)). ⊂ Given a Riemannian manifold (M, ), we have a distance function dist induced by the Riemannian metric.G Gromov showed the following G isoperimetric inequality in L∞(M), which is a generalization to Federer and Fleming’s isoperimetric inequality in Euclidean space, see [7] or [10] for a reference. Theorem 4.4 (Gromov [7]). Let M be a closed Riemannian manifold. For any n-dimensional cycle z L∞(M), ∈ n+1 FillVol(z) 6 Cn Vol(z) n , where Cn is a constant only depending on the cycle dimension n. SYSTOLIC VOLUME AND COMPLEXITY OF 3-MANIFOLDS 9
5. Cubical complexes and δ-extension In this section, we show a tool which is important in the construction of good balls. We will use the cover of such open good balls to construct nerve, see Section 6 in the following. Materials contained in this section are from Gromov [7, Section 6.2.]. A cubical complex is similar to the simplicial complex. The differ- ence is that we take cubes instead of simplexes. Given a manifold M with distance function distM , we define a cubic complex in terms of Kuratowski embedding. Let δ > 0 be a given constant. The standard δ-cube in L∞(M) is the set of functions ϕ(x) 0 6 ϕ(x) 6 δ, for all x M. { | ∈ } Definition 5.1. For a given δ > 0, we define a cubical δ-complex to be a metric space K, which is partitioned into isometric image of standard δ-cubes, such that any two cubes meet at a face. In a cubical δ-complex, there holds the following isoperimetric in- equality. Theorem 5.2 (Gromov [7], Section 6.2.B.). For a given integer n = 1, , there exist two positive constants µn and Cn′ , such that every singular··· n-dimensional cycle z in any cubical δ-complex K with n Vol(z) 6 µnδ bounds a chain c K, and ⊂ (n+1)/n Vol(c) 6 Cn′ [Vol(z)] , (5.1) 1/n such that c is contained in the ε′-neighborhood of z for ε′ = Cn′ (Vol z) .
Assume that M0 M is a finite subset with N elements. The Banach ⊂ N space L∞(M0) is isometric to an N-dimensional space ℓ , where the norm on ℓN is ∞ ∞ (x1, x2, , xN ) = sup xi . k ··· k∞ 16i6N | |
We define a map I : M L∞(M ) as follows, 0 → 0 x ϕ 7→ x with ϕ (w) = min δ, dist (x, w) for any w M . x { M } ∈ 0 The finite dimensional Banach space L∞(M0) can be decomposed into δ-cubes by using the following hyperplanes ϕ ϕ(x)= mδ, x M , { | ∈ 0} 10 L. CHEN where m N are integers. For each point x L∞(M0), after specifying a δ-complex∈ structure, there is a unique cube∈ containing x, denoted (x). Assume that K L∞(M ). For a sufficiently small number ε> ⊂ 0 0, we define a Lipschitz map Rε : K K with the Lipschitz constant 1/( 1 δ ε), which contracts each ε-neighborhood→ of each subcomplex 2 − K0 K onto K0. The map Rε can be defined in terms of the 1- dimensional⊂ Lipschitz map r : [0, 1] [0, 1]: ε → R (x , x , , x )=(r (x ),r (x ), ,r (x )). ε 1 2 ··· N ε 1 ε 2 ··· ε N Definition 5.3. A δ-extension of M, denoted by Kδ(M), is the union
x M (x), ∪ ∈ where (x) stands for the unique δ-cube with minimal dimension and containing the point x′ = R I (x). ε ◦ 0 Let the map J : M K (M) be defined as J = R I . → δ ε ◦ 0 Proposition 5.4. The image M ′ = J(M) of the map J is homeomor- phic to M.
Proof. The finite dimensional approximation K0 : M L∞(M0) of the Kuratowski embedding is bi-Lipschitz, see [1, Proposition→ 5.1.], so that is an embedding. By a slight modification of the proof there, we can see that the map I0 : M L∞(M0) is a local embedding. Since we let ε 6 δ, and each δ-neighborhood→ of point x M is contractible, so that ∈ image Rε(I0(M)) of the Lipschitz map Rε is homeomorphic to I0(M). Hence J is an embedding. δ Note that the map J is Lipchitz with a Lipschitz constant δ 2ε . And for the volume, we have − δ n Vol(M ′) 6 Vol(M). (5.2) δ 2ε − The volume of M ′ is defined in terms of the norm in finite Banach space L∞(M0), see Gromov [7, Section 4]. 6. Good cover and nerve In this section, we introduce the notion of nerve of cover of open balls. The estimate of using nerve of covers can be found in Guth [12] and Gromov [7, Section 5 and Section 6]. Let X be a topological space.
Definition 6.1. The nerve N( ) of an open cover = Uλ λ Λ is a simplicial complex, which satisfies:U U { } ∈ (1) there is a vertex v for each open subset U ; λ λ ⊂ U SYSTOLIC VOLUME AND COMPLEXITY OF 3-MANIFOLDS 11
(2) there is a k-simplex spanned by vertices v , v , , v , if the λ1 λ2 ··· λk corresponding subsets have nonempty intersection, Uλ1 Uλ2 U = . ∩ ∩ ···∩ λk 6 ∅ Denote the underlying polyhedron of N( ) by N. U Definition 6.2. An open cover of X is called a good cover, if each subset U is contractible. U ∈ U Theorem 6.3 (Nerve Theorem, Hatcher [13] Corollary 4G.3.). If X is a paracompact topological space, and N is the underlying polyhedron of the nerve N( ) of a good cover on X, then N is homotopy equivalent to X. U U Given a closed n-dimensional manifold M. Let be an open good cover on M. Then the underlying polyhedron N Uof the nerve N( ) is homotopy equivalent to M, according to the above nerve theorem.U From the manifold M to the underlying polyhedron N of the nerve N( ), there exists canonical Alexandroff map A : M N( ), see KaroubiU and Weibel [20, Proposition 2.1.] or references→ therein.U As- sume that ai is a partition of unity subordinate to the open cover , then the Alexandroff{ } map A takes each point x M to a point withU ∈ barycentric coordinate ai(x) N. Moreover, if K is the underlying∈ polyhedron of a triangulation of M, the Alexandroff map from the manifold to the nerve can be made into simplicial. Let be an open good cover of open balls over K. Moreover, assume thatU each vertex of the underlying polyhedron K is the center of some open ball in the cover . Hence the Alexandroff map is simplicial, if every ball B withU center a vertex of K does not contain any other vertices of K∈. U We take ϕ to be the Alexandroff map from K to N. Lemma 6.4. For the underlying polyhedron K of each triangulation of M, there exists a nerve N( ) of an open good cover of M, whose underlying polyhedron N has moreU vertices than K. U Proof. As mentioned above, there exists a simplicial map ϕ : K N for the nerve N( ) of the open good cover over K. Furthermore,→ with U U respect to different vertices v1, v2 K, the images ϕ(v1) and ϕ2(v2) are distinct due to our previous assumption.∈ Hence in the underlying polyhedron N of the nerve N( ), there will be more vertices than K. U For a closed Riemannian n-manifold (M, ), there exists an open good cover = B (r ) , where B (r ) is anG open ball with center x U { xi i } xi i i and radius ri. 12 L. CHEN
Definition 6.5. Let R = 1 Sys π (M, ). A ball B (r) with radius 0 10 1 G x 0
7. Proof of the main theorem In this section, we prove the main theorem of this paper (Theorem 1.2). The proof is separated into two steps. 7.1. Volume of balls. Let M be a closed n-dimensional aspherical manifold. There exists a sequences of Riemannian metrics k k∞=1, so that {G } Vol k (M) lim G = SR(M). k Sys π (M, )n →∞ 1 Gk Normalize the metric k by scaling to let Sys π1(M, k) = 1, so that we have G G
lim Vol k (M)= SR(M). k G →∞ Denote by Mk the Riemannian manifold (M, k). Let δ > 0 be a given small positive number. We choose a sufficientlyG dense finite subset M ◦ M . For example, we can choose M ◦ to be an ε-net. For k ⊂ k k each k, there is a δ-extension K (M ) L∞(M ◦), see Section 5. The δ k ⊂ k Riemannian manifold can be embedded into Kδ(Mk) by the Lipschitz map J described in Section 5. In order to specify the distance distk SYSTOLIC VOLUME AND COMPLEXITY OF 3-MANIFOLDS 13 induced by the Riemannian metric k on Mk, we use the notation J : M K (M ) to denote the LipschitzG embedding. k k → δ k Let M ′ be the image of M ′ under the Lipschitz map J : M k k k k → Kδ(Mk). Recall that the length and the volume in the Banach space L∞(M) is defined in terms of the norm, see Section 4 or Gromov [7]. The volume of Mk′ satisfies the inequality (5.2). For the purpose of convenience in the following argument, we vary the Riemannian metric on M to let the following fact holds, Gk lim Vol(Mk′ )= SR(M). k →∞ Lemma 7.1. When k is sufficiently large, the volume of every ball
Bw(R) in Mk′ satisfies n Vol(Bw(R)) > AnR , (7.1) where An is a constant only depending on the dimension n. Proof. We prove this Lemma by choosing a minimizing sequence of cycles. Such technique is used by Gromov in [7, Section 4.3. C′′], also see [7, Section 6.4.B.]. Since the δ-extension Kδ(Mk) is compact, we can assume that there is a subsequence of M ′ with limit M of Gromov-Hausdorff conver- { k} ′ gence. For convenience, we still denote the subsequence by M ′ . We f k may further assume that M ′ is minimal, that is, no proper subset of M is the limit of any other minimizing sequence. The limit M of ′ f ′ Gromov-Hausdorff convergence here serves for a formal device of the f f minimizing sequence, see Gromov [7, 4.3.C′′] In the cubic complex Kδ(Mk′ ), there holds the isoperimetric inequal- ity (5.1), see Section 5. Hence by applying an argument analogous to Gromov [7, Theorem 4.3.C′′.], we have n Vol(Bw(R)) > AnR holds for every w M ′ when k is sufficiently large. ∈ k 7.2. Triangulation and nerve of open cover. Assume that k is large enough, so that the estimate (7.1) holds. We construct a nerve
N( ) for an open cover of Mk′ . U 1 U Let R = Sys π (M ′ ). On M ′ K (M ), we choose a system of 0 10 1 k k ⊂ δ k disjoint admissible balls = B (r ) m , see Section 5 for a reference U { xj j }j=1 of admissible balls. Moreover, assume that the system is maximal, and satisfying: U m (1) = B (2r ) is an open cover of M ′ ; U { xj j }j=1 k 14 L. CHEN
(2) each open ball Bxj (2rj) does not contain any other open balls in . U Since the open cover is a good cover, the nerve N( ) is homotopy U U equivalent to Mk′ , so that homotopy equivalent to the manifold M. Given a smooth triangulation of the manifold M, denote by K the underlying polyhedron. In construction of admissible balls in , we U assume that for each vertex v K, the corresponding point in Mk′ is chosen to be the center of an admissible∈ ball. Then we have a simplicial map ϕ : K N, where N is the underlying polyhedron of the nerve N( ). Therefore→ we proved the following result. U
Proposition 7.2. For each smooth triangulation of an aspherical manifold M, there exists an open cover so that theT underlying polyhe- dron of the associated nerve has more vertices than the triangulation . T
Now we show an estimate with respect to the systolic volume. Let the systolic volume be denoted by S in the following.
Proposition 7.3. For each closed aspherical n-manifold M, there ex- ists an open cover , so that the following inequality holds for the number T of simplicesU in the associated nerve and the systolic volume S,
T 6 Bn S exp (Bn′ log S ), (7.2) p where Bn and Bn′ are two constants only depending on the dimension n.
Proof. The covering argument in the proof is the same as in Gromov [7, Theorem 5.3.B.]. Another important reference is Gromov [7, Section 6.4.C.].
Denote by Vk the volume Vol(Mk′ ). Let w Mk′ and Bw(R) Mk′ be an admissible ball. Choose the radius R so∈ that ⊂
k k+1 5− R0 6 R 6 5− R0, SYSTOLIC VOLUME AND COMPLEXITY OF 3-MANIFOLDS 15
1 where k > 1 is an integer. Recall that R0 = 10 Sys π1(Mk′ ). Since we have
1 V > Vol B Sys π (M ′ ) k w 2 1 k
= Vol(Bw(5R0))
> α Vol(Bw(R0)) k 1 k+1 = α Vol B 5 − 5− R w · 0 . . k k+1 > α Vol(Bw(5− R0)) k > α Vol(Bw(R)) k n > α AnR n k n > α5− AnR0 , so that
log V log Rn log A k 6 k − 0 − n . log α n −
Moreover, we choose the constant α to let
log α n = log Sk log An, (7.3) − p −
Vk where we use Sk to denote the systolic ratio n . Let k be the greatest R0 integer satisfying the equality (7.3). Let B (R ) m be a maximal system of mutually disjoint admis- { wj j }j=1 sible open balls, with R (0, R ] and satisfying j ∈ 0
(1) R1 > R2 > > Rm; ··· m (2) = B (2R ) is an open cover of M ′ . U { wj j }j=1 k
Let N be the nerve associated to this cover. Denote by mj the number of balls intersecting with Bwj (Rj). Denote by P the number 16 L. CHEN of pairwise intersections in the open cover . Then we have U Vk = Vol(Mk′ ) m
> Vol(Bwj (Rj)) Xj=1 m 1 > α− Vol(Bwj (5Rj)) Xj=1 m mj 1 > α− Vol(Bwℓ (Rℓ)) Xj=1 Xℓ=1 m mj 1 n > α− AnRℓ Xj=1 Xℓ=1 1 k n > α− An 5− R0 P.
Hence we have
Vk 1 nk P 6 n α An− 5 R0 n 1 (n+1)√log Sk log An =5 An− Sk 5 −
6 Bn Sk exp Bn′ log Sk , p n 1 where Bn = 5 An− , Bn′ = (n + 1) log 5 are constants only depending on n. Denote by Ti the number of i-simplices in the underlying polyhedron N of the nerve N( ), so that T = P , and T = T . Since T 6 2P , U 2 i i 0 and Ti 6 P with i > 2, the estimate (7.2) alsoP holds for any Ti with i > 0, thus holds for T . If we let k , then the estimate (7.2) is obtained. →∞ The inequality (7.2) implies that T SR(M) > Cn , (7.4) exp Cn′ √log T where Cn and Cn′ are two positive constants only depending on the dimension n. Let M be a closed aspherical 3-manifold. In terms of Proposition 7.2, for each triangulation of M, there exists an open cover from max- imal system of disjoint admissible open balls, so that the number t SYSTOLIC VOLUME AND COMPLEXITY OF 3-MANIFOLDS 17 of tetrahedra in this triangulation is no more than the number T3 of tetrahedra in the nerve. Hence we have t SR(M) > C , exp C′ √log t where C and C′ are two positive constants. After taking infimum on t, we get the lower bound estimate of systolic volume in Theorem 1.2.
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