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Systolic Volume and Complexity of 3-Manifolds SYSTOLIC VOLUME AND COMPLEXITY OF 3-MANIFOLDS LIZHI CHEN Abstract. In this paper, we prove that the systolic volume of a closed aspherical 3-manifold is bounded below in terms of com- plexity. Systolic volume is defined as the optimal constant in a systolic inequality. Babenko showed that the systolic volume is a homotopy invariant. Moreover, Gromov proved that the sys- tolic volume depends on topology of the manifold. More precisely, Gromov proved that the systolic volume is related to some topolog- ical invariants measuring complicatedness. In this paper, we work along Gromov’s spirit to show that systolic volume of 3-manifolds is related to complexity. The complexity of 3-manifolds is the min- imum number of tetrahedra in a triangulation, which is a natural tool to evaluate the combinatorial complicatedness. Contents 1. Introduction 2 2. Systolic volume of aspherical manifolds 4 3. Complexity of 3-manifolds 6 4. Kuratowski embedding and isoperimetric inequality 7 5. Cubical complexes and δ-extension 9 6. Good cover and nerve 10 7. Proofofthemaintheorem 12 arXiv:1509.07647v4 [math.GT] 15 Oct 2019 7.1. Volume of balls 12 7.2. Triangulation and nerve of open cover 13 References 17 Date: October 16, 2019. 2010 Mathematics Subject Classification. Primary 53C23, Secondary 57M27. Key words and phrases. Systolic volume, complexity, aspherical 3-manifolds. The work is supported by the Fundamental Research Funds lzujbky-2017-26 for the Central Universities. 1 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.
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