TOPOLOGICAL COMPLEXITY OF CERTAIN ...

TOPOLOGICAL COMPLEXITY OF CERTAIN MOMENT-ANGLE COMPLEXES YASUHIKO KAMIYAMA Abstract. We study the Lusternik-Schnirelmann category and the topological complexity of moment-angle complexes which arise naturally from the motion planning problem.

1. Introduction A moment-angle complex is a very important and efficient tool in toric topology. Thanks to this, one can study various combinatorial objects (like Stanley-Reisner rings, subspace arrangements, cubical complexes, and so on) by methods of algebraic topology. A famous theorem says that the homology of a moment-angle complex admits a natural decomposition. (See, for example, [2, 3]). Recently, lead by two motivations, the homological decomposition was generalized to the space-level decomposition. (See Theorem 2 (iii).) The first motivation is to generalize the homological decomposition to that in any homology theory (see [1]). The second motivation is to study the configuration of a motion planning problem in robotics. More precisely, our robot is the arachnoid mechanism which originated in [12]. It is shown in [9] that the configuration space of an arachnoid mechanism has a model which is a moment-angle complex. (See §2.) The purpose of this paper is to study the Lusternik-Schnirelmann category and the topological complexity of certain moment-angle complexes. The motivation for the study is as follows: As we mentioned above, moment-angle complexes arise naturally from motion planning problem. Recently, the topological complexity TC(X) of a space X is an interest in robotics, because this is the measure of discontinuity of any motion planner in X. (See, for example, [4].) However, computing 2000 Mathematics Subject Classification. Primary 55R80, Secondary 55M30, 68T40. Key words and phrases. moment-angle complex, Lusternik-Schnirelmann category, topological complexity. 1

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YASUHIKO KAMIYAMA

TC(X) for a given X can be an extremely difficult task. For example, TC(RP n ) for n ̸= 1, 3, 7 equals one plus the smallest dimension of Euclidean space into which RP n immerses (see [5]). Let Q(n) denote a regular n-gon in R2 ⊂ R3 . We denote by P1 (n) and P2 (n) the pyramid and prism over Q(n), respectively. The results in [2] show that Q(n) is a manifold of dimension n + 2 and there are homeomorphisms (1)

ZP1 (n) ∼ = Σ2 ZQ(n)

and ZP2 (n) ∼ = S 3 × ZQ(n) .

For a topological space X, let cat(X) denote the Lusternik-Schnirelmann category in the sense of [7] and we set cat0 (X) := cat(XQ ), where XQ denotes the rationalization of X. Moreover, we denote by TC(X) the topological complexity (see [4]) and we set TC0 (X) := TC(XQ ). Our main results are then: Theorem A . We have (i) cat(ZP1 (n) ) = cat0 (ZP1 (n) ) = 2. (ii) TC(ZP1 (n) ) = TC0 (ZP1 (n) ) = 3. Theorem B . We have the following table. Table 1. cat(ZQ(n) ) and cat0 (ZQ(n) ) PP n PP PP 2 3 4 5 6 PP P

cat(ZQ(n) ) cat0 (ZQ(n) )

7

8

2 2 3 3 3 3∼4 3∼4 2 2 3 3 3 3 3

Here 3 ∼ 4 means that cat(ZQ(n) ) is 3 or 4. Theorem C . We have the following tables. Table 2. TC(ZQ(n) ) and TC0 (ZQ(n) ) PP n PP PP 2 3 4 PP P

TC(ZQ(n) ) TC0 (ZQ(n) )

5

6

7

8

3 2 3 4∼5 5 4∼7 5∼7 3 2 3 4 5 4 5

TOPOLOGICAL COMPLEXITY OF CERTAIN MOMENT-ANGLE COMPLEXES3

Table 3. TC(ZP2 (n) ) and TC0 (ZP2 (n) ) PP n PP PP 2 3 4 PP P

TC(ZP2 (n) ) TC0 (ZP2 (n) )

5

6

7

8

4 3 4 5∼6 6 5∼8 6∼8 4 3 4 5 6 5 6

Remark 1. The author does not know why TC0 (ZQ(n) ) does not increase as n. That is, it seems intuitively that the motion planning problem becomes more complicated as the number of legs increases, where we recall the definition of the arachnoid mechanism in §2. 2. Proofs of Theorems A, B and C First, we recall how moment-angle complex arises as a configuration space of robotics. Let a > 0 be a positive real number and K a convex polyhedron in R3 , with vertices v1 , . . . , vn . We assume that ∥v1 ∥ = · · · = ∥vn ∥. We set l(K) = ∥v1 ∥ (= ∥vi ∥, ∀i) and L(K) = min{k1 , . . . , kn }, where we define ki = max1≤j≤n ∥vi − vj ∥. The arachnoid mechanism we consider is a parallel robot in R3 having n two-joined legs, with all joints of a fixed length a/2, joined together at a central point, with the other end of the i-th leg fixed at vi . (See [12].) We define the configuration space M(K, a) ⊂ (R3 )n+1 of such an arachnoid mechanism as follows: M(K, a) := {(p1 , . . . , pn , q) ∥pi − vi ∥ = ∥q − pi ∥ = a/2 (1 ≤ i ≤ n)} . Thus, a point (p1 , . . . , pn , q) ∈ M(K, a) stands for the robot with central point q and internal joint for the i-th leg pi . We denote by ZP the moment-angle complex of a convex polyhedron P . More precisely, let F = {F1 , . . . , Fn } be the set of facets of P . For any facet Fi ∈ F we denote by TFi the one-dimensional coordinate subgroup of T F ∼ = T n that corresponds to Fi . To every face G we assign the coordinate subtorus ⊕ TG = TFi ⊂ T F. Fi ⊃G

For any point q ∈ P , we denote by G(q) the only face that contains q in the relative interior. Then we introduce the quotient space ZP = (T F × P )/ ∼, where (t1 , q) ∼ (t2 , q) if and only if t1 t−1 2 ∈ TG(q) .

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YASUHIKO KAMIYAMA

Consider the case that P is the dual polyhedron of K. Then it is proved in [9] that if l(K) < a < L(K), then there is a homeomorphism M(K, a) ∼ = ZP . In particular, the topological type of M(K, a) is constant for l(K) < a < L(K). Theorems A, B and C are proved by combining known results about ZP , cat(X) and TC(X). We summarize them below. Theorem 2. (i) [2, Theorem 2.15]. For any P , we have π1 (ZP ) = π2 (ZP ) = 0. (ii) [2, 4.4]. If i + j < n + 2, then the cup product map ˜ i (ZQ(n) ) ⊗ H ˜ j (ZQ(n) ) → H ˜ i+j (ZQ(n) ) ∪:H is zero. (iii) [1, 9]. For any P , ΣZP is homotopically equivalent to a bouquet of spheres. In particular, it is proved in [8] that   n−2 ∨ ∨  S i+2  ∨ S n+3 ΣZQ(n) ≅ i=2

α(n,i)

( ) (n) where α(n, i) := n n−2 − i . i−1

Next, for a path-connected topological space X, we denote by cl(X) the cup-length of X. A close relative of cl(X) was defined in [4] as follows: The cup product map (2)

∪ : H ∗ (X) ⊗ H ∗ (X) → H ∗ (X)

is an algebra homomorphism, whose kernel is called the ideal of zerodivisors. The multiplicative structure on the left in (2) is given by the formula (α ⊗ β)(γ ⊗ δ) = (−1)|β||γ| αγ ⊗ βδ. We define the zero-divisors cup-length of X to be the length of the longest non-trivial product of elements in the ideal of zero-divisors. We denote by zcl(X) the zerodivisors cup-length of X. We use the following results. Theorem 3. (i) [7, (1.3)]. cl(X) < cat0 (X) ≤ cat(X). (ii) [7, Proposition 5.1]. If X is (r − 1)-connected (r ≥ 2), then dim X cat(X) ≤ 1 + . r (iii) [4, Theorem 7]. zcl(X) < TC0 (X) ≤ TC(X). (iv) [4, Theorem 5]. TC(X) ≤ 2 · cat(X) − 1. (v) [4, Theorem 11]. TC(X × Y ) ≤ TC(X) + TC(Y ) − 1.

TOPOLOGICAL COMPLEXITY OF CERTAIN MOMENT-ANGLE COMPLEXES5

(vi) [11, Theorem]. If X is an (r − 1)-connected compact manifold of dimension less than or equal to 4r − 2 (where r ≥ 2), then X is a formal space. (vii) [6, p.388, Example 4]. If X is a simply connected formal space, then cat0 (X) = cl(X) + 1. (viii) [10, Theorem 1.2]. If X is a simply connected formal space, then TC0 (X) = zcl(X) + 1. Proof of Theorem A. By (1) and Theorem 2 (iii), ZP1 (n) is homotopically equivalent to a bouquet of spheres. Theorem A is clear from this. ¤ Proof of Theorem B. Since ZQ(n) for n = 2, 3 and 4 are homeomorphic to S 4 , S 5 and (S 3 )2 respectively (see [2]), the results for these n hold. Hence, we may assume that 5 ≤ n ≤ 8. About cat(ZQ(n) ), an upper bound is obtained if we combine Theorem 2 (i) and Theorem 3 (ii). About a lower bound, Theorem 2 (ii) implies that cl(ZQ(n) ) = 2, hence a lower bound is 3 by Theorem 3 (i). About cat0 (ZQ(n) ), Theorem 2 (i) and Theorem 3 (vi) tells us that ZQ(n) is a formal space for n ≤ 8. Then we obtain the results from Theorem 2 (ii) and Theorem 3 (vii). ¤ Proof of Theorem C. Proof of Table 2. As in the proof of Theorem B, we may assume that 5 ≤ n ≤ 8. An upper bound for TC(ZQ(n) ) is given by Theorem 3 (iv). A lower bound is given by computing zcl(ZQ(n) ). For example, the case for n = 6 is computed as follows: By Poincar´e duality, there exist x ∈ H 3 (ZQ(6) ), α, β ∈ H 4 (ZQ(6) ) and y ∈ H 5 (ZQ(6) ) such that xy ̸= 0 and αβ ̸= 0. Then (1 ⊗ x − x ⊗ 1)(1 ⊗ y − y ⊗ 1)(1 ⊗ α − α ⊗ 1)(1 ⊗ β − β ⊗ β) = 2xy ⊗ αβ ̸= 0. Then Theorem 3 (iii) tells us that 5 ≤ TC(ZQ(6) ). Using Theorem 3 (viii), we can compute TC0 (ZQ(n) ) in the same way as in cat0 (ZQ(n) ). ¤ Proof of Table 3. If we use (1), then we can prove Table 3 in the same way as in Table 2. For that purpose, we note that ZP2 (n) is a formal space for n ≤ 8. Indeed, according to [6, p.163, Exercise 2], the product of formal spaces are formal. ¤

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YASUHIKO KAMIYAMA

References [1] A. Bahri, M. Bendersky, F. R. Cohen and S. Gitler, The polyhedral product functor: a method of computation for moment-angle complexes, arrangements and related spaces, preprint, arXiv: 0711.4689. [2] V. M. Buchstaber and T. E. Panov, Torus actions and combinatorics of polytopes, Trudy Mat. Inst. Stekelov. 225 (1999), 96–131; English transl., Proc. Stekelov Inst. Math. 225 (1999), 87–120. [3] V. M. Buchstaber and T. E. Panov, Actions of tori, combinatorial topology and homological algebra, Russian Math. Surveys 55 (2000), 825–921. [4] M. Farber, Topological complexity of motion planning, Discrete Comput. Geom. 29 (2003), 211–221. [5] M. Farber, S. Tabachnikov and S. Yuzvinsky, Topological robotics, motion planning in projective spaces, Int. Math. Res. Not. 34 (2003), 1853–1870. [6] Y. F´elix, S. Halperin and J.-C. Thomas, Rational Homotopy Theory, Graduate Texts in Math. 205, Springer-Verlag, New York, 2001. [7] I. M. James, On category, in the sense of Lusternik-Schnirelmann, Topology 17 (1978), 331–348. [8] Y. Kamiyama and S. Tsukuda, The configuration space of the n-arms machine in the Euclidean space, Topology Appl. 154 (2007), 1447–1464. [9] Y. Kamiyama and S. Tsukuda, On the homology of configuration spaces of arachnoid mechanisms, Houston J. Math. 34 (2008), 483–499. [10] L. Lechuga and A. Murillo, Topological complexity of formal spaces, Topology and Robotics, Contemp. Math. 438, Amer. Math. Soc., Providence, RI, 2007, 105–114. [11] T. J. Miller, On the formality of (k − 1)-connected compact manifolds of dimension less than or equal to 4k − 2, Illinois J. Math. 23 (1979), 253–258. [12] N. Shvalb, M. Shoham and D. Blanc, The configuration space of arachnoid mechanisms, Forum Math. 17 (2005), 1033–1042. Department of Mathematics, University of the Ryukyus, NishiharaCho, Okinawa 903-0213, Japan E-mail address: [email protected]