Linear motion planning with controlled collisions and pure planar braids
aa r X i v : . [ m a t h . A T ] M a r Linear motion planning with controlled collisions andpure planar braids
Jes´us Gonz´alez, Jos´e Luis Le´on-Medinaand Christopher Roque
Contents k ( R , n ) . . . . . . . . . . . . . . 6 TC(Conf k ( R , n ))
94 The higher topological complexity of
Conf k ( R , n )
185 Motion planners for pure planar braids with few strands 196 Configuration spaces with controlled collisions 21
Abstract
We compute the Lusternik-Schnirelmann category (LS-cat) and the higher topo-logical complexity (TC s , s ≥
2) of the “no- k -equal” configuration space Conf k ( R , n ).This yields (with k = 3) the LS-cat and the higher topological complexity of Kho-vanov’s group PP n of pure planar braids on n strands, which is an R -analogue ofArtin’s classical pure braid group on n strands [21]. Our methods can be used todescribe optimal motion planners for PP n provided n is small.2010 Mathematics Subject Classification: 55R80, 55S40, 55M30, 68T40.Keywords and phrases: Motion planning, higher topological complexity, sectional category,configuration spaces, controlled collisions, pure planar braids. Introduction
For a topological space X and a positive integer n , the configuration spaces Conf( X, n ) = { ( x , . . . , x n ) ∈ X n : x i = x j for i = j } , of n ordered points in X , and UConf( X, n ),the orbit space of Conf(
X, n ) by the canonical action of the n -th permutation group, arecentral objects of study in pure and applied mathematics. The case X = C is historicallyand theoretically important: both Conf( C , n ) and UConf( C , n ) are Eilenberg-MacLanespaces of respective types (P n ,
1) and (B n , n stands for Artin’s classical braidgroup on n strands, and P n denotes the corresponding subgroup of pure braids.Having contractible path components, Conf( R , n ) and UConf( R , n ) are topologicallyuninteresting. A meaningful and rich R -analogue of C -based configuration spaces ariseswhen the actual definition of a configuration space is relaxed.For X and n as above, and for an integer k ≥
2, the “no- k -equal” (ordered) config-uration space Conf k ( X, n ) is the subspace of the product X n consisting of the n -tuples( x , . . . , x n ) for which no set { x i , . . . , x i k } , with i j = i ℓ for j = ℓ , is a singleton. Thecorresponding unordered analogue UConf k ( X, n ) is the orbit space of Conf k ( X, n ) by thecanonical action of the n -th permutation group. As shown in [20], the homotopy propertiesof Conf k ( X, n ) (UConf k ( X, n )) interpolate between those of the usual configuration spaceConf(
X, n ) = Conf ( X, n ) (UConf(
X, n ) = UConf ( X, n )), and those of the cartesian(symmetric) n -th power X n = Conf k ( X, n ) (SP n X = UConf k ( X, n )), for k > n . More-over, as discussed in [8], no- k -equal configuration spaces play a subtle role in the study ofthe limit of Goodwillie’s tower of a space of no k -self-intersecting immersions.For the particular case k = 3, Conf ( R , n ) gives the desired R -analogue of the classicalArtin pure braid group. In [21], Khovanov introduces PP n which stands for the group ofplanar pure braids on n strands, also called pure twin group, and proves that Conf ( R , n )is an aspherical space which classifies PP n -principal bundles. No- k -equal configurationspaces on the real line were first considered in [4], where methods for estimating the sizeand depth of decision trees are applied to the analysis of the complexity of the problem ofdetermining whether, for given n real numbers, some k of them are equal.A central goal of this paper is the computation of Farber’s topological complexity (TC)of Conf k ( R , n ) for k ≥
3. In the process, we compute the Lusternik-Schnirelmann category(cat) and all the higher topological complexities (TC s , s ≥
2) of Conf k ( R , n ). Theorem 1.1.
The Lusternik-Schnirelmann category and the topological complexity of
Conf k ( R , n ) are given by cat (Conf k ( R , n )) = ⌊ n/k ⌋ , the integral part of n/k , and TC(Conf k ( R , n )) = , n < k ;1 , n = k with k odd ;2 , n = k with k even ;2 ⌊ n/k ⌋ , n > k. (1)2ee Corollary 4.1 for the corresponding description of all the higher topological com-plexities of Conf k ( R , n ). Note that TC(Conf k ( R , n )) = 2 ⌊ n/k ⌋ unless n = k = 2 ℓ + 1 forsome ℓ > R d , n ) described in [15]:TC(Conf( R d , n )) = n − , d even;2 n − , d odd. (2)Firstly, both (1) and (2) are linear functions on n , of slope 2 in the case of (2), and sloperoughly 2 /k (1 /k if n = k = 2 ℓ + 1) in the case of (1). Further, just as in (2), (1) is atmost one from maximal possible; (2) is precisely one less than maximal possible for d even,while (1) is so only for n = k , an odd number.Since TC( X ) is a homotopy invariant of X , the topological complexity of a group G can be defined as that of any of its classifying spaces, just as in the case of the Lusternik-Schnirelman category cat( G ). In the short but influential paper [10], Eilenberg and Ganealaid the grounds for establishing the fact that cat( G ) agrees with the projective dimensionof the trivial Z [ G ]-module Z . On the other hand, a description of TC( G ) depending solelyon the algebraic properties of G is an open problem which has captured much of the currentattention of the experts in the field. Corollary 1.2.
The category and the topological complexity of PP n are given by cat(PP n ) = ⌊ n/ ⌋ , TC(PP n ) = , n < , n = 3;2 ⌊ n/ ⌋ , n > . Remark 1.3.
J. Mostovoy pointed out to the authors that the cartesian product of ⌊ n/ ⌋ copies of PP sits inside PP n (by cabling sets of 3-strands). In particular, PP n is hyperboliconly for n = 3 , ,
5. In this respect, it is relevant to observe that, while the main resultin [16] asserts that the topological complexity of a hyperbolic group π must be cdim( π × π ) − δ π with δ π ∈ { , } , PP seems to be the only known hyperbolic group π with δ π = 1.Cases with n ≤ and PP are trivial groups, whereas PP , PP and PP are free groupsof respective ranks 1, 7, 31. (The assertion for PP appears as Conjecture 3.5 in [1].)The fact that PP n is free for 3 ≤ n ≤ n = 6. Adirect computation (verifiable using the computational algebraic system GAP) using theReidemeister-Schreier process reveals a group isomorphism PP ∼ = H ∗ F , where F is afree group of rank at least 45. Details of such a fact, as well as potential extensions for3roups PP n with n ≥
6, are the topic of the forthcoming paper [22]. Here we remark that,in any decomposition PP n ∼ = H n ∗ F with F free, the cat and TC values of H n are forcedto agree with those of PP n . Corollary 1.4.
Assume a group isomorphism PP n ∼ = H n ∗ F holds for n ≥ with F afree group. Then cat( H n ) = ⌊ n/ ⌋ and TC( H n ) = 2 ⌊ n/ ⌋ .Proof. This is an immediate consequence of Corollary 1.2 and the formulaecat( G ∗ G ) = max { cat( G ) , cat( G ) } , TC( G ∗ G s ) = max { TC( G ) , TC( G ) , cat( G × G ) } for the free product G ∗ G of arbitrary groups G and G (the TC formula has recentlybeen proved in [9]).Theorem 1.1 has potential applications to current technological developments. For in-stance, Conf k ( R , n ) is the state space of a system consisting of n distinguishable pointsmoving on an interval, and subject to the restriction that k -multiple collisions are forbid-den. For practical applications it is convenient to replace points by intervals of a fixed(suitably small) radius, changing the no- k -multiple-collision condition by the requirementthat no k intervals have a common overlapping. Indeed, it is known (see [8]) that theconfiguration space based on intervals is homotopic to the one based on points. In thiscontext, if the moving objects are equipped with communication sensors, and the radiusof the intervals are thought of as the communication range of each of the moving objects,then the no- k -multiple-collision condition corresponds to the requirement that at most k − k -equal configuration spaces are gen-eralized in the short final Section 6, where we introduce configuration spaces Conf K ( X, n )with collisions controlled by a simplicial complex K . A driving motivation (that arosefrom a lecture of [17, Section 2.4]) is that such spaces (with X = Γ a graph) would seemto be a natural space of states for problems in digital microfluidics (see [11, 12]). In suchprocesses, manipulation of droplets embedded on an inert oil suspension is performed bysuitable application of currents through a grid of wires (the graph Γ) in order to propeldroplets through the wires (due to dynamic surface tension effects). In such a setting, mo-tion planning with controlled collisions (encoded by the complex K ) corresponds to specificmixing process instructions: droplets of various chemical or biological agents would be po-sitioned, mixed, split, and directed to final outputs, all in parallel —an efficient “lab on achip”.It should also be remarked that configuration spaces Conf K ( X, n ) with collisions con-trolled by a simplicial complex K are also interesting outside applications. We thankVictor Turchin for bringing to our attention that these spaces (with X = R ) were usedin [7] to study the homology of the loop space on a polyhedral product ( X , . . . , X m ) K when each space X i is simply connected. 4 Preliminaries
For a space X , the Lusternick-Schnirelmann category, cat( X ), and the topological com-plexity, TC( X ), both homotopy invariants of X , are special cases of the notion of sectionalcategory (or Schwarz genus) of a fibration. Recall that the (reduced) sectional category ofa fibration p : E → B , secat( p ), is defined as the smallest non-negative integer k so thatthere exists an open covering of the base B = U ∪ U ∪ · · · ∪ U k such that the fibration p admits a continuous section on each U i , see [24] . As a special case, we obtain the (re-duced) Lusternick-Schnirelmann category of a space X , cat( X ), defined as the sectionalcategory of the fibration e : P ( X ) → X , where P ( X ) is the space of based paths on X and e is the evaluation map given by e ( γ ) = γ (1). On the other hand, the (reduced)topological complexity of a space X , TC( X ), is defined as the sectional category of thefibration e , : P ( X ) → X × X , where P ( X ) is the space of free paths on X and e , isthe double evaluation map given by e , ( γ ) = ( γ (0) , γ (1)). The open sets U i covering X × X so that e , admits a continuous section on each U i are called local domains, andthe corresponding local sections are called local rules. The system of local domains andlocal rules is called a motion planner for X . A motion planner is said to be optimal if it hasTC( X ) local rules. As explained by Farber in his seminal work [13, 14], this concept givesa homotopical framework for studying the motion planning problem in robotics. Indeed,TC( X ) gives a measure of the complexity of motion-planning an autonomous system withstate-space X and which should perform robustly within a noisy environment.Most of the existing methods to estimate the topological complexity of a given spaceare cohomological in nature and are based on some form of obstruction theory. One of themost (simple and) successful such methods is: Proposition 2.1.
Let X be a c -connected space X having the homotopy type of a CWcomplex, then cl( X ) ≤ cat( X ) ≤ hdim( X ) / ( c + 1) and zcl( X ) ≤ TC( X ) ≤ X ) . The notation hdim( X ) stands for the (cellular) homotopy dimension of X , i.e. theminimal dimension of CW complexes having the homotopy type of X . On the other hand,the cup-length of X , cl( X ), and the zero-divisor cup-length of X , zcl( X ), are defined inpurely cohomological terms. The former one is the largest non-negative integer ℓ such thatthere are coefficients systems A , . . . , A ℓ over X and corresponding positive-dimensionalclasses c j ∈ H ∗ ( X ; A j ) so that the product c · · · c ℓ ∈ H ∗ ( X ; N i A i ) is non-zero. Like-wise, zcl( X ) is the largest non-negative integer ℓ such that there are coefficients systems Schwarz’ original (unreduced) definition is recovered as genus( p ) = secat( p ) + 1. For practical purposes, the openness condition on local domains can be replaced (without altering theresulting numerical value of TC( X )) by the requirement that local domains are pairwise disjoint Euclideanneighborhood retracts (ENR). , . . . , A ℓ over X × X and corresponding classes z j ∈ H ∗ ( X × X ; A j ), each with triv-ial restriction under the diagonal inclusion ∆ : X ֒ → X × X , and so that the product z · · · z ℓ ∈ H ∗ ( X × X ; N i A i ) is non-zero. Each such class z i is called a zero-divisor for X . Throughout this work, we will only be concerned with simple coefficients in Z , andwill omit reference of coefficients in writing a cohomology group H ∗ ( X ). In these terms,∆ ∗ : H ∗ ( X × X ) = H ∗ ( X ) ⊗ H ∗ ( X ) → H ∗ ( X ) is given by cup-multiplication, whichexplains the name “zero-divisors”.All definitions and results reviewed in this subsection have corresponding analogues forRudyak’s higher topological complexity, see [3, 23] for details. Conf k ( R , n ) We recall the description of the cohomology ring H ∗ (Conf k ( R , n )) —see [2, 8]. A binaryrelation R on a set S is any subset of the cartesian product S × S . As usual, we write xRy as a substitute for ( x, y ) ∈ R . A preorder is a binary relation (cid:22) on S which is reflexive( x (cid:22) x, ∀ x ∈ S ) and transitive ( x (cid:22) y (cid:22) z ⇒ x (cid:22) z, ∀ x, y, z ∈ S ). For instance, thediagonal ∆ S = { ( x, x ) : x ∈ S } and the entire cartesian product S × S are preorders whichare called empty and full, respectively. Whenever the preorder is understood, a situationwhere x (cid:22) y and y (cid:22) x is denoted by x ≈ y . Thus, a partial order is a preorder where x ≈ y occurs only with x = y . We write x ≺ y when both x (cid:22) y and x = y hold.The transitive closure of a binary relation R on S is the smallest transitive binaryrelation on S containing R . In particular, the transitive closure of (the union of) twopreorders is automatically a preorder. This yields a commutative and associative binaryoperation ◦ : P ( S ) × P ( S ) → P ( S ) on the set P ( S ) of preorders on S having the emptypreorder as a two-sided neutral element.Fix positive integers n and k with 3 ≤ k ≤ n . Baryshnikov describes the cohomologyring H ∗ (Conf k ( R , n )) in terms of what he calls string preorders, i.e., preorders which are almost determined by a “height” function. Explicitly, a preorder (cid:22) on [ n ] is string if thereis a preorder-preserving map h : [ n ] → R (where R is equipped with the standard order)satisfying x ≺ y whenever h ( x ) < h ( y ), and in such a way that the restriction of (cid:22) to each“level” set h − ( r ) ( r ∈ R ) is either the empty preorder or the full preorder (the heightfunction would fully recover the string preorder if the former would remember which levelsets are empty and which are full). Thus, a string preorder (cid:22) can be spelled out throughthe ordered list (or string) of non-empty level sets of a corresponding height function for (cid:22) ,where the list is ordered increasingly from left to right according to the height values, andenclosing each level subset I ⊆ [ n ] within either [ ]-brackets, if the restriction of (cid:22) to I isfull, or ()-brackets, if the restriction of (cid:22) to I is empty. By convention, a level set with asingle element has to be enclosed within ()-brackets. Recall we only take mod 2 coefficients. In [2], level sets are ordered decreasingly from left to right; this difference is immaterial. elementary , if it has the form ( I )[ J ]( K ) with card( J ) = k − admissible , if it has the form ( I )[ J ]( I )[ J ] · · · [ J d ]( I d ) with card( J i ) = k − i = 1 , . . . d . In such a case, the admissible string preorder is said to have di-mension ( k − d . Elementary string preorders are thus admissible and have dimen-sion k − basic , if it is specified by a string ( I )[ J ]( I )[ J ] · · · [ J d ]( I d ) satisfying card( J i ) = k − J i ∪ I i ) ∈ I i , for all i = 1 , . . . , d (the maximal element of J i ∪ I i is takenwith respect to the standard order of integers). Remark 2.2.
An admissible (basic) preorder ( I )[ J ]( I )[ J ] · · · [ J d ]( I d ) of dimension ( k − d factors as ε ◦ · · · ◦ ε d , where ε i = (cid:16) I ∪ J ∪ I ∪ · · · ∪ J i − ∪ I i − (cid:17) h J i i (cid:16) I i ∪ J i +1 ∪ I i +1 ∪ · · · ∪ J d ∪ I d (cid:17) is an elementary (basic) preorder of dimension k −
2. As a partial converse, note that,for string preorders ( I )[ J ]( K ) and ( I ′ )[ J ′ ]( K ′ ) (possibly non-elementary), the condition I ∪ J ⊆ I ′ implies the equality (cid:16) I (cid:17) h J i (cid:16) K (cid:17) ◦ (cid:16) I ′ (cid:17) h J ′ i (cid:16) K ′ (cid:17) = (cid:16) I (cid:17) h J i (cid:16) K ∩ I ′ (cid:17) h J ′ i ( K ′ ) . (3)(If K ∩ I ′ = ∅ , the ()-level set K ∩ I ′ must be suppressed from the right term in (3).)Likewise, the condition I ′ ∪ J ′ ⊆ I implies the equality (cid:16) I (cid:17) h J i (cid:16) K (cid:17) ◦ (cid:16) I ′ (cid:17) h J ′ i (cid:16) K ′ (cid:17) = (cid:16) I ′ (cid:17) h J ′ i (cid:16) K ′ ∩ I (cid:17) h J i ( K ) . (4)(Correspondingly, if K ′ ∩ I = ∅ , the ()-level set K ′ ∩ I must be suppressed from the rightterm of (4).) The apparent symmetry on the right of (3) and (4) corresponds to the factthat the condition I ∪ J ⊆ I ′ ( I ′ ∪ J ′ ⊆ I ) is equivalent, by complementing, to the condition J ′ ∪ K ′ ⊆ K ( J ∪ K ⊆ K ′ ).The fact that the products in (3) and (4) are string does not depend on the assumedinclusions I ∪ J ⊆ I ′ and I ′ ∪ J ′ ⊆ I . Such a property is not explicitly mentioned (but iscertainly used) in the original works [2, 8]. We include proof details for completeness. Lemma 2.3.
The product of two string preorders ( I )[ J ]( K ) and ( I ′ )[ J ′ ]( K ′ ) is string. Inparticular, if neither the inclusion I ∪ J ⊆ I ′ nor the inclusion I ′ ∪ J ′ ⊆ I hold, then (cid:16) I (cid:17) h J i (cid:16) K (cid:17) ◦ (cid:16) I ′ (cid:17) h J ′ i (cid:16) K ′ (cid:17) = (cid:16) I ∩ I ′ (cid:17) h J ∪ J ′ ∪ ( I ∩ K ′ ) ∪ ( I ′ ∩ K ) i (cid:16) K ∩ K ′ (cid:17) . (5)7 roof. Let (cid:22) stand for the product preorder ( I )[ J ]( K ) ◦ ( I ′ )[ J ′ ]( K ′ ), and for subsets A and B of [ n ] write A (cid:22) B ( A ≈ B ) whenever a (cid:22) b ( a ≈ b ) for all ( a, b ) ∈ A × B . Forinstance, I (cid:22) J (cid:22) K as well as I ′ (cid:22) J ′ (cid:22) K ′ . Since[ n ] = ( I ∩ I ′ ) a (cid:16) J ∪ J ′ ∪ ( I ∩ K ′ ) ∪ ( I ′ ∩ K ) (cid:17) a ( K ∩ K ′ )is clearly a partition, it suffices to show that J ≈ J ′ ≈ ( I ∩ K ′ ) ≈ ( I ′ ∩ K ).Pick x ∈ ( I ∪ J ) \ I ′ and x ′ ∈ ( I ′ ∪ J ′ ) \ I . For any ( j, j ′ ) ∈ J × J ′ , we have • ( x ′ I ⇒ x ′ ∈ J ∪ K ⇒ j (cid:22) x ′ ) and ( x ′ ∈ I ′ ∪ J ′ ⇒ x ′ (cid:22) j ′ ), thus j (cid:22) j ′ ; • ( x I ′ ⇒ x ∈ J ′ ∪ K ′ ⇒ j ′ (cid:22) x ) and ( x ∈ I ∪ J ⇒ x (cid:22) j ), thus j ′ (cid:22) j .Therefore J ≈ J ′ . The result follows from ( I ∩ K ′ (cid:22) J ≈ J ′ (cid:22) K ′ ⇒ I ∩ K ′ ≈ J ≈ J ′ )and ( I ′ ∩ K (cid:22) J ′ ≈ J (cid:22) K ⇒ I ′ ∩ K ≈ J ′ ≈ J ). Corollary 2.4.
Assume ( I )[ J ]( K ) and ( I ′ )[ J ′ ]( K ′ ) are elementary preorders with I ∪ J * I ′ and I ′ ∪ J ′ * I . Then the product in (5): • has the form ( I ′′ )[ J ′′ ]( K ′′ ) with card( J ′′ ) ≥ k − . • is elementary if and only if the preorders ( I )[ J ]( K ) and ( I ′ )[ J ′ ]( K ′ ) agree.Proof. Note that (5) is elementary if and only if J = J ′ and I ∩ K ′ = ∅ = I ′ ∩ K . In sucha case: • I ∩ K ′ = ∅ ⇒ I ⊆ I ′ ∪ J ′ = I ′ ∪ J ⇒ I ⊆ I ′ ; • I ′ ∩ K = ∅ ⇒ I ′ ⊆ I ∪ J = I ∪ J ′ ⇒ I ′ ⊆ I ; • I ∩ K ′ = ∅ ⇒ K ′ ⊆ J ∪ K = J ′ ∪ K ⇒ K ′ ⊆ K ; • I ′ ∩ K = ∅ ⇒ K ⊆ J ′ ∪ K ′ = J ∪ K ′ ⇒ K ⊆ K ′ .So in fact I = I ′ and K = K ′ .We are now ready to state Baryshnikov’s description of the ring H ∗ (Conf k ( R , n )).Recall we are assuming Z coefficients. Theorem 2.5 (Baryshnikov [2, Theorem 1], Dobrinskaya-Turchin [8, Section 4]) . For k ≥ , the cohomology ring H ∗ (Conf k ( R , n )) is isomorphic to the (anti)commutative freeexterior algebra generated in dimension k − by the elementary preorders subject to thefollowing relations:1. P a ∈ A ( A \ { a } ) h { a } ∪ B i ( C ) = P c ∈ C ( A ) h B ∪ { c } i ( C \ { c } ) , whenever [ n ] can bewritten as a disjoint union [ n ] = A ` B ` C with card( B ) = k − . . ( I )[ J ]( K ) · ( I ′ )[ J ′ ]( K ′ ) = 0 , for elementary preorders ( I )[ J ]( K ) and ( I ′ )[ J ′ ]( K ′ ) whose transitive closure ( I )[ J ]( K ) ◦ ( I ′ )[ J ′ ]( K ′ ) has a [ ] -level set of cardinality largerthan k − . Remark 2.6.
Since H ∗ (Conf k ( R , n )) is a quotient of an exterior algebra, Remark 2.2,Lemma 2.3 and Corollary 2.4 imply that a (cup) product ( I )[ J ]( K ) · ( I ′ )[ J ′ ]( K ′ ) of twoelementary preorders of dimension 1, ( I )[ J ]( K ) and ( I ′ )[ J ′ ]( K ′ ), can (potentially) be non-zero only when the (transitive-closure) product ( I )[ J ]( K ) ◦ ( I ′ )[ J ′ ]( K ′ ) is of dimension 2.Further, the latter condition holds precisely when one (and necessarily only one) of theinclusions I ∪ J ⊆ I ′ and I ′ ∪ J ′ ⊆ I holds, in which case the (transitive closure) product( I )[ J ]( K ) ◦ ( I ′ )[ J ′ ]( K ′ ) is given by (3) and (4), respectively.Going one step further, Baryshnikov shows that the difference between cup productsand transitive-closure products can safely be neglected: Theorem 2.7 (Baryshnikov [2, Theorem 2], Dobrinskaya-Turchin [8, Section 4]) . Addi-tively, H ∗ (Conf k ( R , n )) is free with (graded) basis given by the cup products of elementarypreorders whose transitive-closure product is basic—as in the first assertion in Remark 2.2. Cup products of elementary preorders whose corresponding transitive-closure productfails to be basic can be written in terms of basic ones by iterated use of the first relationin Theorem 2.5. The process is clarified in the next section, where we work extensivelyin terms of Baryshnikov’s basis in H ∗ (Conf k ( R , n )), and the corresponding tensor basis in H ∗ (Conf k ( R , n ) × Conf k ( R , n )) ∼ = H ∗ (Conf k ( R , n )) ⊗ H ∗ (Conf k ( R , n )). TC(Conf k ( R , n )) Theorem 1.1 is obvious for n ≤ k . In fact, for n < k , Conf k ( R , n ) = R n , which iscontractible, so that TC(Conf k ( R , n )) = 0. On the other hand, for n = k and with∆ = { ( x, x, . . . , x ) : x ∈ R } , Conf k ( R , n ) = R k − ∆ ≃ S k − , whose topological complexityis well known to be 1 (respectively 2) if k is odd (respectively even). We thus assume n > k in what follows (recall we also assume k ≥ k ( R , n ) (and thus the assertion in Theorem 1.1 about the latter number) are easilyestablished: Lemma 3.1.
Conf k ( R , n ) is a ( k − )-connected space having cat(Conf k ( R , n )) = ⌊ n/k ⌋ and hdim(Conf k ( R , n )) = ( k − ⌊ n/k ⌋ . In particular, for k = 3 , both the cohomologicaldimension ( cdim ) and the geometric dimension ( gdim ) of the group PP n equal ⌊ n/k ⌋ .Proof. Let q = ⌊ n/k ⌋ . The Baryshnikov basis element h , . . . , k − i (cid:16) k (cid:17) h k + 1 , . . . , k − i (cid:16) k (cid:17) · · · h ( q − k + 1 , . . . , qk − i (cid:16) qk (cid:17) (6)9s a (non-zero) product of q factors, each being a dimension-1 basis element (see the firstassertion in Remark 2.2), which implies q ≤ cat(Conf k ( R , n )). On the other hand, [25,Theorems 1.1 and 1.2] imply that Conf k ( R , n ) is ( k − k − k − q . The first two assertionsin the lemma then follow from the inequality cat ≤ (hdim) / (conn +1) —which in turnfollows from a standard obstruction-theory argument. The last assertion in the lemma(for k = 3, so hdim(Conf k ( R , n )) = gdim(PP n ), by definition), follows from the relationscat = cdim ≤ gdim in [10].We have omitted the use of curly braces for level sets within the string preorder (6).This convention will be kept throughout the rest of the paper.The standard inequality TC( X ) ≤ X ) yields TC(Conf k ( R , n )) ≤ ⌊ n/k ⌋ . Thus,in view of Proposition 2.1, the proof of Theorem 1.1 will be complete once we show2 ⌊ n/k ⌋ ≤ zcl(Conf k ( R , n )) , for n > k ≥
3. (7)In order to establish (7), we introduce a few key elements in H ∗ (Conf k ( R , n )) and in H ∗ (Conf k ( R , n )) ⊗ . (Recall that all cohomology groups will be taken with Z -coefficients,a restriction that is not essential but allows us to simplify calculations.) Definition 3.2.
For a positive integer m satisfying m + k ≤ n + 2 , consider the elements x m , x ′ m ∈ H k − (Conf k ( R , n )) given by x m = (cid:16) , . . . , m − , m − (cid:17) h m, m + 1 , . . . , m + k − i (cid:16) m + k − , . . . , n (cid:17) ,x ′ m = (cid:16) , . . . , m − , m (cid:17) h m − , m + 1 , . . . , m + k − i (cid:16) m + k − , . . . , n (cid:17) , where x ′ m is defined only for m ≥ . Each of the corresponding zero-divisor y m = x m ⊗ ⊗ x m for Conf k ( R , n ) is central in what follows, with the elements x ′ m playing a subtlerole. Note that x m and x ′ m are Baryshnikov basis elements in H ∗ (Conf k ( R , n )) provided m + k ≤ n + 1. In fact, as illustrated by the first assertion in Remark 2.2, i Y j =1 x ( j − k +2 = x x k +2 · · · x ( i − k +2 (8)= (cid:16) (cid:17)h , . . . , k i(cid:16) k + 1 (cid:17)h k + 2 , . . . , k i(cid:16) k + 1 (cid:17) · · · h ( i − k + 2 , . . . , ik i(cid:16) ik + 1 , . . . , n (cid:17) is a basis element in H ∗ (Conf k ( R , n )) provided ik + 1 ≤ n . Likewise, if e x ( j − k +1 standsfor either x ( j − k +1 or x ′ ( j − k +1 (the latter one being a possibility only for j ≥ i Y j =1 e x ( j − k +1 = e x e x k +1 · · · e x ( i − k +1 (9)= h , · · · , k − i(cid:16) k ❑ ✕ (cid:17)h k + 1 , · · · , k − i · · · (cid:16) ( i − k ■ ✒ (cid:17)h ( i − k + 1 , · · · , ik − i(cid:16) ik, . . . , n (cid:17) , e x ( j − k +1 under consideration), is a basis element in H ∗ (Conf k ( R , n ))provided ik ≤ n . Example 3.3.
The condition 3 ≤ k < n ensures that both x and x are Baryshnikovbasis elements in H ∗ (Conf k ( R , n )), and since x = x , we obviously have y y = ( x ⊗ ⊗ x )( x ⊗ ⊗ x ) = · · · + x ⊗ x + x ⊗ x + · · · 6 = 0 . (10)So 2 ≤ zcl(Conf k ( R , n )), which readily yields (7) for 2 k > n > k ≥ n ≥ k and k ≥ y y on the left-hand side of (10). Most importantly, the tensor factors x and x in thetwo highlighted summands on the right-hand side of (10) will be replaced by products ofthe form (8), and by certain products of the form (9), some of which are made explicit asfollows: p i, = x (cid:16)Q a − j =1 x (2 j − k +1 x ′ jk +1 (cid:17) x (2 a − k +1 , if i = 2 a ≥ x (cid:16)Q aj =1 x (2 j − k +1 x ′ jk +1 (cid:17) , if i = 2 a + 1 ≥ p i, = x (cid:16)Q a − j =1 x ′ (2 j − k +1 x jk +1 (cid:17) x ′ (2 a − k +1 , if i = 2 a ≥ x (cid:16)Q aj =1 x ′ (2 j − k +1 x jk +1 (cid:17) , if i = 2 a + 1 ≥ Theorem 3.4.
If the integers i, k, n satisfy ≤ i , ≤ k , and ik ≤ n , then the product i Y j =1 y ( j − k +1 y ( j − k +2 ∈ H ∗ (Conf k ( R , n )) ⊗ (11) is non-zero. Explicitly:1. If ik + 1 ≤ n , then the expression of (11) as a linear combination of Baryshnikovtensor basis elements for H ∗ (Conf k ( R , n )) ⊗ uses the Baryshnikov basis element i Y j =1 x ( j − k +1 ⊗ i Y j =1 x ( j − k +2 .
2. If ki = n , then the expression of (11) as a linear combination of Baryshnikov tensorbasis elements for H ∗ (Conf k ( R , n )) ⊗ uses the Baryshnikov basis element p i, ⊗ p i, . As distilled in Example 3.3, the hypothesis i ≥ n = k , for which y y is forced to vanish (recall the Z -coefficients!) in view of the first paragraph of this11ection. (By working over the integers, rather than over Z , the (truly!) exceptional casewould only be reduced to that where n = k is odd.)The validness of (7) for n ≥ k and k ≥ i = ⌊ n/k ⌋ . So, the rest of the section is devotedto the proof of Theorem 3.4. Lemma 3.5.
The following relations hold in H ∗ (Conf k ( R , n )): x x k +1 = x x k +1 , for n ≥ k − .2. x n − k +4 x n − k +2 = x n − k +3 x n − k +2 = 0 , for n ≥ k − .3. x n − k +2 x n − k +2 = x n − k +2 x n − k +1 , for n ≥ k − .4. x n − k +1 x n − k +2 = x n − k +1 x n − k +1 + x n − k +1 x ′ n − k +1 , for n ≥ k .5. x r x r + k x r +2 k − = x r x r + k − x r +2 k − , for n ≥ r + 3 k − and r ≥ .6. x r x r + k +1 x r +2 k = x r x r + k x r +2 k + x r x ′ r + k x r +2 k , for n ≥ r + 3 k − and r ≥ . Remark 3.6.
The numeric restrictions on k , n and r ensure that each of the factors x m in the six items above is an element of H ∗ (Conf k ( R , n )). Proof of Lemma 3.5.
All these equalities follow from Theorem 2.5 and Remark 2.6. Wegive full details for completeness.Assume n ≥ k −
2. Take A = { , . . . , n − k + 1 } , B = { n − k + 2 , . . . , n − } and C = { n } in Theorem 2.5.1 to get x n − k +2 = (1 , . . . , n − k + 1)[ n − k + 2 , . . . , n ]= n − k +1 X i =1 (1 , . . . , b i, . . . , n − k + 1)[ i, n − k + 2 , . . . , n − n ) . (12)As explained in Remark 2.6, all terms in the summation in (12) vanish when multipliedby x n − k +3 = (1 , . . . , n − k + 2)[ n − k + 3 , . . . , n − k + 1]( n − k + 2 , · · · , n ). Thisyields x n − k +3 x n − k +2 = 0, while the equality x n − k +4 x n − k +2 = 0 follows directly from theconsiderations in Remark 2.6. This proves item 2.Assume n ≥ k −
1. Terms with i ≤ n − k in the summation in (12) vanish whenmultiplied by x n − k +2 = (1 , . . . , n − k + 1)[ n − k + 2 , . . . , n − k ]( n − k + 1 , . . . , n ). Thisyields x n − k +2 x n − k +2 = x n − k +2 x n − k +1 , proving item 3.Assume n ≥ k . Terms with i < n − k in the summation in (12) vanish when mul-tiplied by x n − k +1 = (1 , . . . , n − k )[ n − k + 1 , . . . , n − k − n − k, . . . , n ). This yields x n − k +1 x n − k +2 = x n − k +1 x n − k +1 + x n − k +1 x ′ n − k +1 , proving item 4.12ssume n ≥ r +3 k − r ≥
1. Take A = { , . . . , r + k − } , B = { r + k, . . . , r +2 k − } and C = { r + 2 k − , . . . , n } in Theorem 2.5.1 to get r + k − X i =1 (1 , . . . , b i, . . . , r + k − i, r + k, . . . , r + 2 k − r + 2 k − , . . . , n )= n X i = r +2 k − (1 , . . . , r + k − r + k, . . . , r + 2 k − , i ]( r + 2 k − , . . . , b i, . . . , n ) . Terms with i < r + k − x r = (1 , . . . , r − r, . . . , r + k − r + k − , . . . , n ), and terms with i > r + 2 k − x r +2 k − = (1 , . . . , r + 2 k − r + 2 k − , . . . , r + 3 k − r + 3 k − , . . . , n ). This yields the equality x r x r + k − x r +2 k − = x r x r + k x r +2 k − , proving item 5.When n ≥ k −
1, the previous argument applies for r = 2 − k —by vacuity in thecase of the assertion about the first summation, whose only one term is x . This yields x x k +1 = x x k +1 , proving item 1.Assume n ≥ r +3 k − r ≥
1. Take A = { , . . . , r + k } , B = { r + k +1 , . . . , r +2 k − } and C = { r + 2 k − , . . . , n } in Theorem 2.5.1 to get r + k X i =1 (1 , . . . , b i, . . . , r + k )[ i, r + k + 1 , . . . , r + 2 k − r + 2 k − , . . . , n )= n X i = r +2 k − (1 , . . . , r + k )[ r + k + 1 , . . . , r + 2 k − , i ]( r + 2 k − , . . . , b i, . . . , n ) . (13)Terms with i < r + k − x r = (1 , . . . , r − r, . . . , r + k − r + k − , . . . , n ), while terms with i > r +2 k − x r +2 k = (1 , . . . , r +2 k − r +2 k, . . . , r +3 k − r +3 k − , . . . , n ).This yields the equality x r x ′ r + k x r +2 k + x r x r + k x r +2 k = x r x r + k +1 x r +2 k , proving item 6. Proof of part 1 in Theorem 3.4.
By Remark 2.6, y ( j − k +1 y ( j − k +2 = ( x ( j − k +1 ⊗ ⊗ x ( j − k +1 )( x ( j − k +2 ⊗ ⊗ x ( j − k +2 )= x ( j − k +1 ⊗ x ( j − k +2 + x ( j − k +2 ⊗ x ( j − k +1 , so the product in (11) is i Y j =1 y ( j − k +1 y ( j − k +2 = ( x ⊗ x + x ⊗ x )( x k +1 ⊗ x k +2 + x k +2 ⊗ x k +1 ) · · ·· · · ( x ( i − k +1 ⊗ x ( i − k +2 + x ( i − k +2 ⊗ x ( i − k +1 )= X ǫ j ∈ { , } ≤ j ≤ i x − ǫ x k +3 − ǫ · · · x ( i − k +3 − ǫ i ⊗ x ǫ x k + ǫ · · · x ( i − k + ǫ i . (14)13he basis element we care about, namely i Y j =1 x ( j − k +1 ⊗ i Y j =1 x ( j − k +2 , (15)is the summand in (14) with ǫ j = 2 for all j . The proof task is to argue that, when weexpand the other terms of (14) as sums of tensor of basis elements, the tensor (15) doesnot appear. This is obvious for the summand in (14) with ǫ j = 1 for all j . For all othersummands, the assertion will be argued by focusing on the sequence of leaps associated tothe subscripts of both tensor factors of each summand in (14). Explicitly, the first leap inthe subscripts of x − ǫ x k +3 − ǫ · · · x ( i − k +3 − ǫ i is k + 3 − ǫ − (3 − ǫ ) = k + ǫ − ǫ , and thefull sequences of leaps associated to x − ǫ x k +3 − ǫ · · · x k ( i − − ǫ i and x ǫ x k + ǫ · · · x k ( i − ǫ i (16)are, respectively,( k + ǫ − ǫ , k + ǫ − ǫ , . . . , k + ǫ i − − ǫ i ) and ( k − ǫ + ǫ , k − ǫ + ǫ , . . . , k − ǫ i − + ǫ i ) . (17)Such sequences of leaps clearly satisfy:(A) Leap values are either k − k , or k + 1. Moreover, if all k -leaps are removed fromeither one of the sequences in (17), then the resulting sequence of leaps either is emptyor, else, has leap values that alternate between k − k + 1: ( k − k + 1, k − k + 1, k − k + 1,. . . ).(B) The two sequences of leaps in (17) are coordinate-wise complementary to each otherwith respect to 2 k .(C) The first leap different from k (if any) in either of the sequences of leaps (17) is a( k + 1)-leap (( k − x ( x ).Since the right tensor factor in (15), i.e. Q ij =1 x ( j − k +2 , is a basic string preorder startingas (1)[2 , . . . , k ] · · · , the proof is complete in view of Proposition 3.7 below. Proposition 3.7.
Any summand in (14) whose associated sequences of leaps (17) containat least a ( k − -leap (equivalently a ( k + 1) -leap) is a linear combination of tensor basiselements u ⊗ v where both u and v are basic string preorders starting as [1 , . . . , k − I ) · · · ( I i − )[ J i ]( I i ) . roof. Take a product p = x k x k · · · x k i in (16), so k ∈ { , } , with associated sequenceof leaps ( ℓ , . . . , ℓ i − ) satisfying conditions (A)–(C) above, and so that not all leap values ℓ j are k . Case k = 1 : p has the form x · · · x kr +1 x k ( r +1)+2 | {z } ( k + 1)-leap · · · x kr +2 x k ( r +1)+1 | {z } ( k − · · · x kr +1 x k ( r +1)+2 | {z } ( k + 1)-leap · · · x kr +2 x k ( r +1)+1 | {z } ( k − · · · , (18)where we only indicate ( k − k + 1)-leaps. Items 5 and 6 in Lemma 3.5allow us to replace each portion x kr j +1 x k ( r j +1)+2 · · · x kr j +1 +2 x k ( r j +1 +1)+1 , having an initial( k + 1)-leap, a final ( k − k -leaps, by x kr j +1 ( x k ( r j +1)+1 + x ′ k ( r j +1)+1 ) x k ( r j +2)+1 · · · x kr j +1 +1 x k ( r j +1 +1)+1 , which only has k -leaps. The replacing process can be iterated since the initial and finalterms in the replacing portion agree with those in the replaced portion. After all replace-ments are made, and sums are distributed, p becomes a sum of expressions each of whichis similar to the original one (18), except that some of the initial x kj +1 ’s get replaced bythe corresponding x ′ kj +1 , and in such a way that no ( k − k + 1)-leap shows up. But any such expression is a basis element of the required form (thelatter assertion uses the hypothesis ik + 1 ≤ n in part 1 of Theorem 3.4 —see Remark 3.8below). Case k = 2 : p has the form x · · · x kr +2 x k ( r +1)+1 | {z } ( k − · · · x kr +1 x k ( r +1)+2 | {z } ( k + 1)-leap · · · x kr +2 x k ( r +1)+1 | {z } ( k − · · · x kr +1 x k ( r +1)+2 | {z } ( k + 1)-leap · · · , Items 1 and 5 in Lemma 3.5 allow us to replace the initial portion x · · · x kr +2 x k ( r +1)+1 by x · · · x kr +1 x k ( r +1)+1 . Then, the replacement process described in the previous caseallows us to write p as a sum of basis elements of the required form. Remark 3.8.
Part 2 in Theorem 3.4 will be proved using an argument similar to that inthe previous proof, except that it will be necessary to deal first with an additional subtlety.Namely, note that when ik = n , we have x ( i − k +2 = (cid:16) , · · · , ( i − k + 1 (cid:17) h ( i − k + 2 , · · · , ik i (cid:16) ik + 1 , · · · , n (cid:17) = (cid:16) , · · · , ( i − k + 1 (cid:17) h ( i − k + 2 , · · · , n i , which is an elementary non-basic element (i.e., under the main hypothesis in part 2 ofTheorem 3.4). So, when analyzing a typical tensor factor x ǫ x k + ǫ · · · x ( i − k + ǫ i in (14) with ǫ i = 2, the recursive process described in the previous proof will not end up producingsums of basis elements. This issue will be resolved using item 4 in Lemma 3.5.15et us go back to the starting point for the proof of part 2 in Theorem 3.4, i.e., theexpression in (14) for the product Q ij =1 y ( j − k +1 y ( j − k +2 . As observed in Remark 3.8, weno longer work with the basis element indicated in part 1 of Theorem 3.4. Instead, thebasis element we now care about is p i, ⊗ p i, , where ki = n , and which arises from one ofthe two summands in (14) for which the values of the indices ǫ j alternate between 1 and2. In order to simplify the argument, it is convenient to note that all y j , and therefore theirproduct Q ij =1 y ( j − k +1 y ( j − k +2 , are invariant under the involution induced by the map thatswitches coordinates in Conf k ( R , n ) × Conf k ( R , n ). We show the following (equivalent, bythe symmetry just noted, but slightly simpler-to-prove) version of part 2 in Theorem 3.4: Theorem 3.9.
For i ≥ , k ≥ and n = ki , both p i, ⊗ p i, and p i, ⊗ p i, are usedin the expression of the product (11) as a linear combination of Baryshnikov tensor basiselements for H ∗ (Conf k ( R , n )) ⊗ .Proof. We provide full proof details when i = 2 a is even; the parallel argument for i oddis left as an exercise for the reader. In order to simplify notation, we let r · r · · · r t and r · r · · · r t | s · s · · · s t stand for x r x r · · · x r t and x r x r · · · x r t ⊗ x s x s · · · x s t , respectively.With this notation, (14) becomes (cid:16) | | (cid:17) (cid:16) ( k + 1) | ( k + 2) + ( k + 2) | ( k + 1) (cid:17) · · ·· · · (cid:16) ((2 a − k + 1) | ((2 a − k + 2) + ((2 a − k + 2) | ((2 a − k + 1) (cid:17) = X ǫ j ∈ { , } ≤ j ≤ i (3 − ǫ )( k + 3 − ǫ ) · · · ((2 a − k + 3 − ǫ a ) (cid:12)(cid:12)(cid:12) ( ǫ )( k + ǫ ) · · · ((2 a − k + ǫ a ) . (19)The summand with ( ǫ , ǫ , · · · , ǫ a ) = (1 , , , . . . ,
2) is2 · ( k + 1) · (2 k + 2) · (3 k + 1) · · · ((2 a − k + 2) · ((2 a − k + 1) (cid:12)(cid:12)(cid:12) · ( k + 2) · (2 k + 1) · (3 k + 2) · · · ((2 a − k + 1) · ((2 a − k + 2) , (20)whose associated sequences of leaps are( k − , k + 1 , k − , . . . , k −
1) and ( k + 1 , k − , k + 1 , . . . , k + 1) . (21)Using the replacing process explained in the previous proof, it is clear that the expression of2 · ( k + 1) · (2 k + 2) · (3 k + 1) · · · ((2 a − k + 2) · ((2 a − k + 1)in terms of Baryshnikov basis elements uses p a, , but not p a, . Likewise, the replacing processand item 4 in Lemma 3.5 imply that the expression of1 · ( k + 2) · (2 k + 1) · (3 k + 2) · · · ((2 a − k + 1) · ((2 a − k + 2) n terms of Baryshnikov basis uses p a, . Therefore the expression of (20) in terms of Baryshnikov(tensor) basis elements uses p a, ⊗ p a, without using p a, ⊗ p a, . Further, the symmetry comingfrom the involution induced by the switching map on Conf k ( R , n ) × implies that the expressionin terms of Baryshnikov basis of the summand in (19) with ( ǫ , ǫ , · · · , ǫ a ) = (2 , , . . . ,
1) uses p a, ⊗ p a, without using p a, ⊗ p a, .It remains to prove that neither p a, ⊗ p a, nor p a, ⊗ p a, are used in the expression interms of basis elements of any summand in (19) whose associated sequences of leaps is differentfrom those in (21). By symmetry, it suffices to consider the case of a summand(3 − ǫ )( k + 3 − ǫ ) · · · ((2 a − k + 3 − ǫ a ) (cid:12)(cid:12)(cid:12) ( ǫ )( k + ǫ ) · · · ((2 a − k + ǫ a ) (22)with ǫ = 1. Let λ ∈ { k − , k, k + 1 } ( ρ ∈ { k + 1 , k, k − } ) stand for the value of the last leapin the tensor factor on the left (right) of (22). Recall λ + ρ = 2 k . Case λ = ρ = k : The ending portion of one of the two tensor factors in (22) is forced to be · · · ((2 a − k + 1) · ((2 a − k + 1) . The replacing process shows that such a factor cannot give rise to p a, or p a, in its expressionin terms of Baryshnikov basis. Case ( λ, ρ ) = ( k − , k + 1) : The equalities ǫ a − = 1 and ǫ a = 2 are now forced. Letting j ′ stand for x ′ j , and ignoring Baryshnikov basis elements different from p a, and p a, , the rightfactor in (22) then becomes1 · ( k + ǫ ) · · · ((2 a − k + 1)((2 a − k + 2) = 1 · ( k + ǫ ) · · · ((2 a − k + 1)((2 a − k + 1) ′ , in view of the replacing process and item 4 in Lemma 3.5. Further, the replacing process makesit clear that the expression of the latter element in terms of Baryshnikov basis elements does notuse p a, , and that it uses p a, only if the sequence of leaps associated to the right tensor factorin (22) is the second sequence in (21). Case ( λ, ρ ) = ( k + 1 , k − : The equalities ǫ a − = 2 and ǫ a = 1 are now forced. IgnoringBaryshnikov basis elements different from p a, and p a, , the left factor in (22) becomes2 · ( k + 3 − ǫ ) · · · ((2 a − k + 1)((2 a − k + 2)= 2 · ( k + 3 − ǫ ) · · · ((2 a − k + 1)((2 a − k + 1) ′ , where the latter expression further evolves under the replacing process (still ignoring Baryshnikovbasis elements different from p a, and p a, ) to either zero or to2 · ( k + 1) · (2 k + 1) · (3 k + 1) ′ · · · ((2 a − k + 1)((2 a − k + 1) ′ . (23)Note the factor “( k + 1)”, rather than a (primed) “( k + 1) ′ ”, due to the initial “2” in (23). In anycase, a final application of item 1 in Lemma 3.5 shows that (23) vanishes modulo Baryshnikovbasis elements different from p a, and p a, . The higher topological complexity of
Conf k ( R , n ) We can now easily deduce the value of the higher topological complexity TC s (Conf k ( R , n )),for any s > Corollary 4.1.
For s > , TC s (Conf k ( R , n )) = , n < k ; s − , n = k with k odd; s, n = k with k even; s ⌊ n/k ⌋ , n > k. Proof.
The case n ≤ k is trivial. For n > k and s >
2, Lemma 3.1 and [3, Theo-rem 3.9] imply the estimate TC s (Conf k ( R , n )) ≤ s ⌊ n/k ⌋ . From [3, Definition 3.8 andTheorem 3.9], equality will follow once we exhibit a non-zero product of s ⌊ n/k ⌋ “ s -thzero-divisors” for Conf k ( R , n ), i.e., of elements in the kernel of the iterated cup product H ∗ (Conf k ( R , n )) ⊗ s → H ∗ (Conf k ( R , n )).Let i = ⌊ n/k ⌋ , q ∈ { , . . . , s − } , and consider the s -th zero-divisors z m,q = 1 ⊗ · · · ⊗ ⊗ x m |{z} q -th ⊗ ⊗ · · · ⊗ ⊗ · · · ⊗ ⊗ x m ∈ H ∗ (Conf k ( R , n )) ⊗ s , whenever m + k ≤ n + 2. For instance i Y j =1 z ( j − k +1 ,s − z ( j − k +2 ,s − = 1 ⊗ · · · ⊗ ⊗ i Y j =1 y ( j − k +1 · y ( j − k +2 and, for q ≤ s − z m,q i Y j =1 z ( j − k +1 ,s − z ( j − k +2 ,s − = 1 ⊗ · · · ⊗ ⊗ x m |{z} q − th ⊗ ⊗ · · · ⊗ ⊗ i Y j =1 y ( j − k +1 · y ( j − k +2 + 1 ⊗ · · · ⊗ ⊗ (1 ⊗ x m ) · i Y j =1 y ( j − k +1 y ( j − k +2 . The second summand in the latter expression vanishes in view of Lemma 3.1 (by dimen-sional considerations or, alternatively, by cat-considerations). Consequently i Y j =1 z ( j − k +1 , · i Y j =1 z ( j − k +1 , · · · i Y j =1 z ( j − k +1 ,s − · i Y j =1 z ( j − k +1 ,s − z ( j − k +2 ,s − = i Y j =1 x ( j − k +1 ⊗ · · · ⊗ i Y j =1 x ( j − k +1 ⊗ i Y j =1 y ( j − k +1 y ( j − k +2 , s − In a recent work ([1]), Bardakov, Singh and Vesnin have proved:(i) PP n is free of rank (1 ,
7) for n = (3 , n is not free for n ≥ is a free group of rank 31.The proof of (i) occupies a full section in [1]. In fact, the authors of that paper offer twodifferent proofs of the freeness of PP , one with a geometric flavor and another one with analgebraic flavor. The algebraic proof is technical, whereas the geometric proof is extensive.In this section we give short elementary arguments for both (i) and (ii), as well as a shortargument proving a stronger form (Proposition 5.1 below) of the conjectured (iii). Inaddition, we indicate a way to construct an explicit optimal motion planner for PP n when n is small.Under this paper’s perspective, the simplest case is that of (ii), which is an immediateconsequence of Corollary 1.2 and the well-known fact that the topological complexity of afree group is at most 2. Even easier is the case n = 3 in (i). Indeed, as observed at thebeginning of Section 3, Conf k ( R , n ) is either contractible or has the homotopy type of thesphere S k − for, respectively, n < k or n = k . In particular PP and PP are trivial, while(and relevant for (i)) PP is an infinite cyclic group.Condition (iii) is a special case of: Proposition 5.1.
For ≤ k < n < k , Conf k ( R , n ) has the homotopy type of a wedge of β ( k, n ) spheres of dimension k − , where β ( k, n ) = Σ ni = k (cid:16) ni (cid:17)(cid:16) i − k − (cid:17) . Proof.
Severs-White have shown in [25, Theorem 1.1] that Conf k ( R , n ) admits a minimalcellular model, i.e., it has the homotopy type of a cell complex having as many cells ineach dimension d as the rank of the homology (free abelian) group H d (Conf k ( R , n )). Theresult then follows from Theorem 2.7 and [5, Theorem 1.1(c)].19e next give an elementary geometric argument leading to a proof of (i) and (iii), as wellas to a description of explicit motion planners for the corresponding groups PP n . Considerthe subspace X n ⊂ Conf ( R , n ) consisting of the elements x = ( x , . . . , x n ) ∈ Conf ( R , n )with x n = 0 and | x | = 1. For instance X = n ( x , x , ∈ R : | ( x , x ) | = 1 and ( x , x ) = (0 , o = S , (24)whereas X = ( x , x , x , ∈ R : | ( x , x , x ) | = 1,(0 ,
6∈ { ( x , x ) , ( x , x ) , ( x , x ) } not all x , x , x are equal = S − {± (0 , , , ± (0 , , , ± (1 , , , ± √ , , } = R − { } ≃ _ S . (25) Lemma 5.2 (Compare to [19, Section III]) . Let R + stand for the positive real num-bers. For n ≥ , the map f : Conf ( R , n ) × R + × R → Conf ( R , n ) sending the triple (( x , . . . , x n ) , r, a ) into ( x r + a, . . . , x n r + a ) yields, by restriction, a homeomorphism X n × R + × R ∼ = Conf ( R , n ) . Consequently, the subspace inclusion X n ֒ → Conf ( R , n ) is a homotopy equivalence.Proof. For the first assertion, it is straightforward to check that the inverse of the restric-tion of f to X n × R + × R is given by the map g : Conf ( R , n ) → X n × R + × R sending( x , . . . , x n ) into the triple (cid:18) N ( x − x n , . . . , x n − − x n , , N, x n (cid:19) , where N stands for the norm of ( x − x n , . . . , x n − − x n , N ∈ R + since( x , . . . x n ) ∈ Conf ( R , n ) and n ≥
3. For the second assertion, note that the composite X n ֒ → Conf ( R , n ) ∼ = X n × R + × R takes the form x ( x, , X as the complement in R of ten unlinked and untangled curves. Details are omitted. Lemma 5.2 can be used to construct optimal motion planners on Conf ( R , n ) for smallvalues of n . Details are based on a couple of reductions using the following standard In private communications, Harshman and Knapp report having also carried out the reductions for X analogous to (24) and (25), and which lead to a geometric verification of the fact that PP is free ofrank 31. α : X → Y is a homotopy equivalence withhomotopy inverse β : Y → X . Fix a homotopy H : X × [0 , → X between H = β ◦ α andthe identity H = Id : X → X . Assume s : U → P ( Y ) is a local rule for Y (i.e. a sectionfor the double evaluation map e , : P ( Y ) → Y × Y ) on the open set U ⊆ Y × Y , and set V = ( α × α ) − ( U ). Then a local rule σ : V → P ( X ) for X is defined through the formula σ ( x , x )( t ) = H ( x , t ) , for 0 ≤ t ≤ / β ( s ( α ( x ) , α ( x ))(3 t − , for 1 / ≤ t ≤ / H ( x , − t )) , for 2 / ≤ t ≤ F = (cid:16) Conf ( R , n ) ∼ = → X n × R + × R proj −→ X n (cid:17) , we see that it suffices to describe an optimal motion planner on X n . (Explicit formulaefor F , the needed homotopy inverse G , and the needed homotopy between the identityand the composite G ◦ F are easily deduced from the proof of Lemma 5.2.) In turn, since X n has the homotopy type of a wedge of circles (for n ≤ X ≃ S , X ≃ ∨ S and X ≃ ∨ S . The latter task has been accomplished in (24) and (25) for n = 3 and n = 4, where an obvious stereographic projection is needed in the latter case.The resulting motion planner in Conf ( R ,
3) is spelled out next.
Example 5.3.
Let D be the subspace of Conf ( R , × Conf ( R ,
3) consisting of pairs( x, y ) such that the line segment [ x, y ] in R from x to y does not intersect the diagonal∆ = { ( z, z, z ) : z ∈ R } , and let D be the complement of D in Conf ( R , × Conf ( R , D and D are ENR’s, so it suffices to describe a local rule on each. Motion planningin Conf ( R ,
3) for points ( x, y ) ∈ D can be done by following the segment [ x, y ]. On theother hand, for ( x, y ) ∈ D , let p ( x, y ) be the point where the segment [ x, y ] intersects ∆.Since the vectors y − x and (1 , ,
1) are linearly independent, their cross product u ( x, y ) isnonzero. We then motion plan in Conf ( R ,
3) from x to y (with ( x, y ) ∈ D ) by followingfirst the segment [ x, p ( x, y ) + u ( x, y )], and then the segment [ p ( x, y ) + u ( x, y ) , y ]. No- k -equal configuration spaces are special instances of configuration spaces with collisionscontrolled by a simplicial complex. Fix a space X , a positive integer n , and a simplicialcomplex K with vertex set { , . . . , n } . Let ∆ n − ,d stand for the d -dimensional skeleton of∆ n − , the simplex spanned by { , . . . , n } .For a subset σ ⊆ { , . . . , n } consider the partial diagonal subspace D σ := { ( x , . . . , x n ) ∈ X n : card( { x i : i ∈ σ } ) = 1 } . efinition 6.1. The K -diagonal in X n is D K = S σ D σ , where σ runs over the non-facesof K . We set Conf K ( X, n ) = X n − D K , which we call the configuration space of n points in X with collisions controlled by K . By definition, for ( x , . . . , x n ) ∈ Conf K ( X, n ), a multiple collision x i = x i = · · · = x i d can hold only if { i , i , . . . , i d } ∈ K .Note that D σ ⊆ D τ provided τ ⊆ σ . Consequently:1. Conf L ( X, n ) ⊆ Conf K ( X, n ), provided L is a subcomplex of K .2. Conf K ( X, n ) = X n − S σ D σ , where σ runs over the minimal non-faces of K .For instance:3. Conf ∆ n − ( X, n ) = X n .4. Conf ∆ n − , ( X, n ) = Conf(
X, n ), the usual configuration space.5. Conf ∆ n − ,k − ( X, n ) = Conf k ( X, n ), the no- k -equal configuration space of X .Although configuration spaces with collisions controlled by simplicial complexes haveappear previously in the literature, most of their algebraic topology properties are presentlyunknown. An interesting challenge for future research is to compute their mod 2 coho-mology algebras, say as modules over the Steenrod algebra, in terms of the combinatorialproperties of the controlling simplicial complexes. As in the case of Conf k ( R , n ), thismight give enough information to compute the topological complexity of Conf K ( R , n ) asa function of n and K . References [1] Valeriy Bardakov, Mahender Singh, and Andrei Vesnin. Structural aspects of twinand pure twin groups. To appear in
Geometriae Dedicata
DOI: 10.1007/s10711-019-00429-1. arXiv:1811.04020 [math.GR].[2] Yu. Baryshnikov. On the cohomology ring of no k -equal manifolds. Preprint 1997.Available from https://publish.illinois.edu/ymb/home/papers/ .[3] Ibai Basabe, Jes´us Gonz´alez, Yuli B. Rudyak, and Dai Tamaki. Higher topologicalcomplexity and its symmetrization. Algebr. Geom. Topol. , 14(4):2103–2124, 2014.[4] Anders Bj¨orner and L´aszl´o Lov´asz. Linear decision trees, subspace arrangements andM¨obius functions.
J. Amer. Math. Soc. , 7(3):677–706, 1994.225] Anders Bj¨orner and Volkmar Welker. The homology of “ k -equal” manifolds andrelated partition lattices. Adv. Math. , 110(2):277–313, 1995.[6] Daniel C. Cohen and Goderdzi Pruidze. Motion planning in tori.
Bull. Lond. Math.Soc. , 40(2):249–262, 2008.[7] Natalia Dobrinskaya. Loops on polyhedral products and diagonal arrangements.Preprint from 2009. Avaliable from arXiv:0901.2871 [math.AT].[8] Natalya Dobrinskaya and Victor Turchin. Homology of non- k -overlapping discs. Ho-mology Homotopy Appl. , 17(2):261–290, 2015.[9] A. Dranishnikov and R. Sadykov. The topological complexity of the free product.To appear in Mathematische Zeitschrift (2018). https://doi.org/10.1007/s00209-018-2206-y.[10] Samuel Eilenberg and Tudor Ganea. On the Lusternik-Schnirelmann category ofabstract groups.
Ann. of Math. (2) , 65:517–518, 1957.[11] R. Fair, A. Khlystov, V. Srinivasan, V. Pamula, and K. Weaver. Integrated chem-ical/biochemical sample collection, pre-concentration, and analysis on a digital mi-crofluidic lab-on-a-chip platform.
In Lab-on-a-Chip: Platforms, Devices, and Appli-cations, Conf. 5591, SPIE Optics East, Philadelphia, Oct. 25?28, 2004 .[12] R. Fair, V. Srinivasan, H. Ren, P. Paik, V. Pamula, and M. Pollack. Electrowettingbased on chip sample processing for integrated microfluidics.
IEEE Inter. ElectronDevices Meeting (IEDM) , 2003.[13] Michael Farber. Topological complexity of motion planning.
Discrete Comput. Geom. ,29(2):211–221, 2003.[14] Michael Farber. Instabilities of robot motion.
Topology Appl. , 140(2-3):245–266, 2004.[15] Michael Farber and Mark Grant. Topological complexity of configuration spaces.
Proc. Amer. Math. Soc. , 137(5):1841–1847, 2009.[16] Michael Farber and Stephan Mescher. On the topological complexity of asphericalspaces. arXiv:1708.06732v2 [math.AT] .[17] Robert Ghrist. Configuration spaces, braids, and robotics. In
Braids , volume 19 of
Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap. , pages 263–304. World Sci. Publ.,Hackensack, NJ, 2010.[18] Jes´us Gonz´alez, B´arbara Guti´errez, and Sergey Yuzvinsky. Higher topological com-plexity of subcomplexes of products of spheres and related polyhedral product spaces.
Topol. Methods Nonlinear Anal. , 48(2):419–451, 2016.2319] N.L Harshman and N.L. Knapp. Anyons from three-body hard-core interactions inone dimension. arXiv:1803.11000 [math-ph] .[20] Sadok Kallel and Ines Saihi. Homotopy groups of diagonal complements.
Algebr.Geom. Topol. , 16(5):2949–2980, 2016.[21] Mikhail Khovanov. Real K ( π,
1) arrangements from finite root systems.
Math. Res.Lett. , 3(2):261–274, 1996.[22] Jacob Mostovoy and Christopher Roque. The planar pure braid group on six strands.
In preparation .[23] Yuli B. Rudyak. On higher analogs of topological complexity.
Topology Appl. ,157(5):916–920, 2010.[24] Schwarz, A. S. The genus of a fiber space.
Amer. Math. Soc. Transl. (2) , 55:49–140,1966.[25] Christopher Severs and Jacob A. White. On the homology of the real complement ofthe k -parabolic subspace arrangement. J. Combin. Theory Ser. A , 119(6):1336–1350,2012.
Departamento de Matem´aticasCentro de Investigaci´on y de Estudios Avanzados del I.P.N.Av. Instituto Polit´ecnico Nacional n´umero 2508San Pedro Zacatenco, M´exico City 07000, M´exico [email protected]@math.cinvestav.mx
Instituto de Matem´aticasUniversidad Nacional Aut´onoma de M´exicoLe´on No.2, Altos, Oaxaca de Ju´arez 68000, M´exico [email protected]@im.unam.mx