Generating functions and topological complexity
aa r X i v : . [ m a t h . A T ] M a r GENERATING FUNCTIONS AND TOPOLOGICALCOMPLEXITY
MICHAEL FARBER, DAISUKE KISHIMOTO, AND DONALD STANLEY
Abstract.
We examine the rationality conjecture raised in [1] whichstates that (a) the formal power series P r ≥ TC r +1 ( X ) · x r representsa rational function of x with a single pole of order 2 at x = 1 and (b)the leading coefficient of the pole equals cat ( X ). Here X is a finite CW-complex and for r ≥ TC r ( X ) denotes its r -th sequentialtopological complexity. We analyse an example (violating the Ganeaconjecture) and conclude that part (b) of the rationality conjecture isfalse in general. Besides, we establish a cohomological version of therationality conjecture. Introduction
For a topological space X the symbol X r stands for the r -th Carte-sian power of X and ∆ r : X → X r denotes the diagonal map ∆ r ( x ) =( x, x, . . . , x ), where x ∈ X . For r ≥ r -th topological complexity of X is an integer TC r ( X ) ≥ r : X → X r . In other words, TC r ( X ) is the smallest k ≥ X r = U ∪ U ∪ · · · ∪ U k with the property thateach set U i admits a homotopy section s i : U i → X of ∆ r , i.e. the composi-tion ∆ r ◦ s i is homotopic to the inclusion U i → X r for i = 0 , , . . . , k . Theabove definition is equivalent to the one given in [3]. For r > TC r ( X ) is sometimes called the higher or sequential topological complexity . TC r ( X ) is a homotopy invariant of X . For a mechanical system having X asits configuration space, TC r ( X ) can be interpreted as complexity of motionplanning algorithms taking r states ( x , x , . . . , x r ) ∈ X r as input and pro-ducing a continuous motion of each of the points x i moving to a consensusstate y ∈ X . See [3], [2], [1] for more detail.Putting all the numbers TC r ( X ) together, we obtain a formal power series F X ( x ) = X r ≥ TC r +1 ( X ) x r , (1) the TC -generating function of X . It was observed in [1] that in many ex-amples this series represents a rational function of a special type. Based on M. Farber was partially supported by a grant from the Leverhulme Foundation.The authors thank Fields Institute for Research in Mathematical Sciences forhospitality. these computations, it was conjectured in [1] that for any finite CW-complex X the power series F X ( x ) is a rational function of the form P X ( x )(1 − x ) (2) where the numerator P X ( x ) is an integer polynomial satisfying P X (1) = cat ( X ) . The conjecture is true for all spheres, closed orientable surfaces,simply connected symplectic manifolds and also for the Eilenberg-MacLanespaces X = K ( H Γ ,
1) where H Γ is the right angled Artin group determinedby a graph Γ, see [1].The goal of this paper is to show that the above conjecture is false inthe form it was stated in [1], namely we show that for a specific finite CW-complex X the power series F X ( x ) is a rational function of the form (2)however the value P X (1) is distinct from cat ( X ).2. Rationality of homological power series
We start with a general observation:
Lemma 1.
Consider a formal power series F ( x ) = P r ≥ t r x r with integercoefficients t r ∈ Z . The following conditions are equivalent: (a) F ( x ) represents a rational function of the form P ( x ) · (1 − x ) − where P ( x ) is an integer polynomial; (b) the coefficients t r satisfy for all r large enough a recurrent equationof the form t r = t r − + a ;(c) for some a and c one has t r − + a ≤ t r ≤ ra + c for all r largeenough.Proof. First note that P r ≥ x r = (1 − x ) − and P r ≥ ( r + 1) x r = (1 − x ) − .Assuming (a), one can write the polynomial P ( x ) in the form P ( x ) = a + a (1 − x ) + (1 − x ) q ( x ) which leads to F ( x ) = a (1 − x ) + a (1 − x ) + q ( x ) , where q ( x ) is an integer polynomial. Equating coefficients, for r large enoughwe obtain t r = a ( r + 1) + a and hence t r = t r − + a with a = a . Hence(a) = ⇒ (b).Obviously (b) = ⇒ (c).Given (c), we may write t r = ra + s r and the LHS inequality in (c)gives s r − ≤ s r while the RHS inequality gives s r ≤ c , for r large. Hencethe integer sequence s r is bounded above and eventually increasing, thus itconverges, and therefore it is eventually constant, i.e. s r = d and t r = ra + d for all r large enough, implying (b).Finally, to show that (b) = ⇒ (a) we note that t r = ra + d for all large r implies that F ( x ) equals the sum of ax (1 − x ) + d − x and an integer polynomialand therefore F ( x ) can be written in form P ( x ) · (1 − x ) − where P ( x ) is aninteger polynomial; note that P (1) = a . (cid:3) ENERATING FUNCTIONS AND TOPOLOGICAL COMPLEXITY 3
Let A be a graded commutative finite dimensionalalgebra with unit over a field k . We shall assume that A is connected , i.e. thedegree zero part A = k is generated by the unit 1 ∈ A . A typical exampleis the case when A = H ∗ ( X, k ) where X is a connected finite CW-complex.We denote by cl ( A ) the cup-length of A , it is the largest number of el-ements of A of positive degree with nonzero product. Equivalently, cl ( A )equals the largest number of homogeneous elements of A of positive degreewith nonzero product.If B is another such algebra one may consider the tensor product A ⊗ B ,the algebra structure is given by ( a ⊗ b ) · ( a ′ ⊗ b ′ ) = ( − | a ′ |·| b | aa ′ ⊗ bb ′ . Here a, a ′ , b, b ′ are homogeneous elements and the symbol | a | denotes the degreeof a .One notes that cl ( A ⊗ B ) = cl ( A ) + cl ( B ) . (3)Indeed, if a , . . . , a r ∈ A and b , . . . , b s ∈ B elements of positive degree with a a . . . a r = 0 ∈ A and b b . . . b s = 0 ∈ B then r Y i =1 a i ⊗ · s Y j =1 ⊗ b j = 0 ∈ A ⊗ B. Conversely, to show that cl ( A ⊗ B ) ≤ cl ( A ) + cl ( B ) consider a nonzeroproduct of homogeneous tensors cl ( A ⊗ B ) Y ℓ =1 a ℓ ⊗ b ℓ = 0 ∈ A ⊗ B. The set of indices { , . . . , cl ( A ⊗ B ) } = C ⊔ C ⊔ C can be split into 3disjoint sets where for ℓ ∈ C one has | b ℓ | = 0 and for ℓ ∈ C one has | a ℓ | = 0 and for ℓ ∈ C one has | a ℓ | 6 = 0 = | b ℓ | . Clearly, | C | + | C | ≤ cl ( A )and | C | + | C | ≤ cl ( B ) implying cl ( A ⊗ B ) = | C | + | C | + | C | ≤ cl ( A )+ cl ( B ).For an integer r ≥ A r the r -fold tensor product A r = A ⊗ A ⊗ · · · ⊗ A ( r factors) . The above discussion gives:
Corollary 1.
For any r > one has cl ( A r ) = r cl ( A ) , and therefore X r ≥ cl ( A r ) · x r = cl ( A ) · x (1 − x ) . As above, let A be a connected gradedcommutative finite dimensional algebra with unit over a field k . For any r ≥ µ r : A r → A where µ r ( a ⊗ a ⊗ · · · ⊗ a r ) = a a · · · a r . (4)This map is a homomorphism of algebras and its kernel has a graded algebrastructure. The kernel ker( µ r ) is called the ideal of zero divisors . MICHAEL FARBER, DAISUKE KISHIMOTO, AND DONALD STANLEY
Definition 1.
The r -th zero-divisors-cup-length of A , denoted zcl r ( A ), isthe longest nontrivial product of elements of ker( µ r ). Lemma 2.
The power series P r ≥ zcl r +1 ( A ) · x r is a rational function ofthe form P ( x ) · (1 − x ) − where P ( x ) is an integer polynomial satisfying P (1) = cl ( A ) .Proof. First we observe that (using (3)), zcl r ( A ) ≤ cl ( A r ) = r · cl ( A ) . (5)Next we claim that zcl r +1 ( A ) ≥ zcl r ( A ) + cl ( A ) . (6)Indeed, suppose that zero-divisors x , . . . , x ℓ ∈ ker( µ r ) are such that theirproduct x · x · · · x ℓ = 0 ∈ ker( µ r ) is nonzero; here ℓ = zcl r ( A ). If ℓ ′ denotes cl ( A ), consider elements of positive degree y , . . . , y ℓ ′ ∈ A with nonzeroproduct y · y · · · y ℓ ′ = 0 ∈ A . For i = 1 , , . . . , ℓ ′ define the elements y i ∈ A r +1 given by y i = 1 ⊗ ⊗ · · · ⊗ y i − y i ⊗ ⊗ · · · ⊗ . Clearly, y i lies inker( µ r +1 ). Besides, let x j = x j ⊗ ∈ A r +1 ; again, we have x j ∈ ker( µ r +1 ).Finally we claim that the productΠ = x · x · · · x ℓ · y · y · · · y ℓ ′ ∈ A r +1 is nonzero. Let ψ : A → k be a linear functional satisfying ψ ( y y . . . y ℓ ′ ) = 1and ψ | A i = 0 for all i distinct from the degree of the product y y . . . y ℓ ′ .Then the map 1 ⊗ ⊗· · ·⊗ ⊗ ψ : A r +1 → A r satisfies (1 ⊗ ⊗· · ·⊗ ⊗ ψ )(Π) = x · x · · · x ℓ = 0 showing that Π = 0. This proves (6).The statement of Lemma 2 now follows (by applying Lemma 1) from (5)and (6). (cid:3) In applications we have A = H ∗ ( X ; k ) is the cohomology algebra of aconnected finite CW-complex X ; then the number zcl r ( A ) serves as thelower bound for TC r ( X ), see [3].We shall abbreviate the notation zcl r ( H ∗ ( X, k )) to zcl r ( X ; k ) or to zcl r ( X )assuming that the field k is specified. Corollary 2.
Let X be a connected finite CW-complex. For any field k , theformal power series P r ≥ zcl r +1 ( X ; k ) · x r represents a rational function ofthe form P ( x ) · (1 − x ) − with P (1) = cl ( H ∗ ( X ; k )) . Theorem 1.
Let X be a connected finite CW-complex such that for alllarge r one has TC r ( X ) = zcl r ( H ∗ ( X ; k )) where k is a field. Then the TC -generating function (1) is a rational function of form P ( x ) · (1 − x ) − with P (1) = cl ( H ∗ ( X ; k )) . Theorem 1 suggests how to produce an examplecontradicting the rationality conjecture as stated in [1]. Namely, supposethat X is a finite CW-complex satisfying TC r ( X ) = zcl r ( X ; k ) for all large r although cl ( H ∗ ( X ; k )) < cat ( X ). Then the TC -generating function (1) is ENERATING FUNCTIONS AND TOPOLOGICAL COMPLEXITY 5 a rational function of form P ( x ) · (1 − x ) − while the leading term P (1) issmaller than cat ( X ) . In section 5 of [4] the author constructs a remarkable finite CW-complex X . In general X depends on a choice of a prime p but for our purposes inthis paper we shall for simplicity assume that p = 3. The complex X has 3cells of dimensions 2, 3, 11. Its main properties are:(i) cat ( X ) = 2;(ii) cat ( X × X ) = 2;(iii) For any non-contractible space Y one has cat ( X × Y ) < cat ( X ) + cat ( Y ) . In particular cat ( X × S n ) = cat ( X ) = 2 for any n ≥
1, i.e. X is a counterex-ample to the Ganea conjecture. Using these properties we find by inductionthat cat ( X r ) ≤ r for any r ≥ A = H ∗ ( X ; k ) of X has generators a , a , a of degrees 2 , ,
11 correspondingly; all pairwise products of the generatorsvanish for dimensional reasons. We see that the cup-length cl ( A ) = cl ( H ∗ ( X ; k )) = 1and using Corollary 1 we obtain cl ( H ∗ ( X r ; k )) = r for any r . Thus combiningwith information given above, we see that for any r ≥ cat ( X r ) = r. (7)Next we show that zcl r ( A ) ≥ r for any r ≥
2, where A = H ∗ ( X ; k ).Consider the class x = a ⊗ − ⊗ a ∈ ker( µ ) ⊂ A . We see that x = − a ⊗ a = 0 which implies zcl ( A ) ≥
2. Applying (6) inductively wededuce zcl r ( A ) ≥ r for any r ≥
2. Therefore, TC r ( X ) ≥ r, for any r ≥ . (8)Combining (7) and (8) with the well-known inequalities zcl r ( X ) ≤ TC r ( X ) ≤ cat ( X r ) we obtain TC r ( X ) = r = zcl r ( X ) for any r ≥ . (9)By Theorem 1 the TC -generating function P r =1 TC r +1 ( X ) · x r is a rationalfunction of the form P ( x ) · (1 − x ) − where P ( x ) is an integer polynomialwith P (1) = 1. Here 1 = cl ( A ) < cat ( X ). Thus this example violates theoriginal rationality conjecture of [1]. The discussion above suggests a weaker version of therationality conjecture which can be stated as follows: for any finite CW-complex X the TC -generating function F X ( x ) = P r ≥ TC r +1 ( X ) · x r is arational function with a single pole of order 2 at x = 1. MICHAEL FARBER, DAISUKE KISHIMOTO, AND DONALD STANLEY
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On the Lusternik-Schnirelmann Category of Maps , Canad. J. Math. Vol. (3), 2002 pp. 608–633. School of Mathematical Sciences, Queen Mary University of London, Lon-don, E1 4NS, United Kingdom
E-mail address : [email protected] Department of Mathematics, Graduate School of Science, Kyoto Univer-sity, Kyoto 606-8502, Japan
E-mail address : [email protected] Department of Mathematics and Statistics, University of Regina, 3737 Was-cana Parkway, Regina, Saskatchewan, Canada
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