Topology of parametrised motion planning algorithms
aa r X i v : . [ m a t h . A T ] S e p TOPOLOGY OF PARAMETRISED MOTION PLANNINGALGORITHMS
DANIEL C. COHEN, MICHAEL FARBER, AND SHMUEL WEINBERGER
Abstract.
In this paper we introduce and study a new concept of parame-terised topological complexity, a topological invariant motivated by the motionplanning problem of robotics. In the parametrised setting, a motion planningalgorithm has high degree of universality and flexibility, it can function undera variety of external conditions (such as positions of the obstacles etc). We ex-plicitly compute the parameterised topological complexity of obstacle-avoidingcollision-free motion of many particles (robots) in 3-dimensional space. Ourresults show that the parameterised topological complexity can be significantlyhigher than the standard (nonparametrised) invariant. Introduction
To create an autonomously functioning system in robotics one designs a motionplanning algorithm. Such an algorithm takes as input the initial and the finalstates of the system and produces a motion of the system from the initial to finalstate, as output. The theory of robot motion planning algorithms is an active fieldof robotics, we refer the reader to the monographs [15], [16] for further references.A topological approach to the robot motion planning problem was developedin [6], [7]; the topological techniques explained relationships between instabilitiesoccurring in robot motion planning algorithms and topological features of robots’configuration spaces.In this paper we develop a new approach to theory of motion planning algo-rithms. We want our algorithms to be universal or flexible in the sense that theyshould be able to function in a variety of situations, involving external conditionswhich are viewed as parameters and are part of the input of the algorithm. Atypical situation of this kind arises when we are dealing with collision free motion Mathematics Subject Classification.
Key words and phrases. parameterised topological complexity, obstacle-avoiding collision-freemotion planning.D. Cohen was partially supported by an LSU Faculty Travel Grant.M. Farber was partially supported by a grant from the Leverhulme Foundation. of many objects (robots) moving in the 3-space avoiding a set of obstacles, andthe positions of the obstacles are a priori unknown. This specific problem servesas the main motivation for us in this article and we analyse it in full detail.To model the situation mathematically we use the language of algebraic topol-ogy. We consider a map p : E → B which allows to viewed E as the union of afamily of fibres X = X b = p − ( b ), parametrised by the points of the base b ∈ B .A choice of a point of the base b ∈ B corresponds to a choice of the external con-ditions for the system. The parametrised topological complexity TC [ p : E → B ]is a reflection of complexity of a universal motion planning algorithm in this set-ting; we also use the abbreviated notation TC B ( X b ) = TC B ( X ) to emphasise itsrelationship to the fibre. We establish several basic results describing the natureof this notion and in particular give lower and upper bounds depending on thetopological spaces involved.Our main result is described in Section 9 where we prove the following Theorem: Theorem 1.1.
The parametrised topological complexity of the problem of collision-free motion of n robots in the 3-space in the presence of m point obstacles withunknown a priori position equals n + m − . This result can be compared with the well-known answer for the standard (i.e.nonparametrised) topological complexity of the problem of collision-free motion of n robots in the presence of m obstacles which equals 2 n , see [9]. Thus we see thatthe parametrised topological complexity can exceed the non-parametrised one andtheir difference can be arbitrarily large, reflecting computational cost for flexiblemotion planning. The result shows that, unlike ordinary topological complexity,each additional obstacle imposes a cost on the motion planner.In this paper we consider robots and obstacles represented by single points whichmay appear unrealistic. However it is well-known, that our approach is equivalent(under cetain conditions) to the situation when every robot and obstacle has convexshape.The main effort of the proof of Theorem 1.1 is in analysis of algebraic propertiesof the cohomology algebra of a corresponding configuration space, see Section 9.Theorem 1.1 is stated for the 3-dimensional case, however our methods prove asimilar result in Euclidean space of any odd dimension. The answer in the even-dimensional case is slightly different and its treatment requires different tools. Weplan to describe the even-dimensional case in a separate publication. We shallalso develop explicit parametrised motion planning algorithms having minimaltopological complexity for a variety of situations important for applications. OPOLOGY OF PARAMETRISED MOTION PLANNING ALGORITHMS 3 Parametrised motion planning
In robotics, a motion planning algorithm takes as input pairs of admissible statesof the system and generates a continuous motion of the system connecting thesetwo states, as output. Let X be the configuration space of the system which isa path-connected topological space. Given a pair of states ( x , x ) ∈ X × X , amotion planning algorithm produces a continuous path γ : I → X with γ (0) = x and γ (1) = x , where I = [0 ,
1] is the unit interval. Let X I denote the space of allcontinuous paths in X (with the compact-open topology). The map π : X I → X × X, π ( γ ) = ( γ (0) , γ (1)) , (2.1)is a fibration, with fiber Ω X , the based loop space of X . A solution of the motionplanning problem, a motion planning algorithm, is then a section of this fibration,i.e. a map s : X × X → X I satisfying π ◦ s = id X × X . The section s cannot becontinuous as a function of the input unless the space X is contractible, see [6].For a path-connected topological space X , the topological complexity TC ( X )is defined to be the sectional category, or Schwarz genus, of the fibration (2.1), TC ( X ) = secat ( π ). That is, TC ( X ) is the smallest number k for which there is anopen cover X × X = U ∪ U ∪ · · · ∪ U k and the map π admits a continuous section s j : U j → X I satisfying π ◦ s j = id U j for each j . Refer to the survey [7] and therecent volume [12] for detailed discussions of the invariant TC ( X ), which providesa measure of navigational complexity in X . Recent important results on TC ( X )were obtained in [2], [13].In this paper, we pursue a parameterised version of topological complexity,where the motion of the system is constrained by conditions imposed by points inanother topological space.To describe the setting of parametrised motion planning consider a fibration p : E → B. (2.2)In general we only require (2.2) to be a Hurewicz fibration but in most applicationsit will be a locally trivial fibration. Here the space B parametrises the externalconditions for the system and for any value b ∈ B the fibre X b = p − ( b ) is the spaceof “real” or “achievable ”configurations of the system under the external condition b . We assume throughout that the fibre X b is nonempty and path-connected forany b ∈ B . A motion planning algorithm takes as input the current and the desiredstates of the system and produces a continuous motion from the current state tothe desired state; the additional conditions in the parametrised setting are: (a)the current and the desired states must lie in the same fibre of p (i.e. they haveto satisfy the same external conditions) and (b) the motion of the system must D. COHEN, M. FARBER, AND S. WEINBERGER also be restricted to the same fibre; in practical terms this additional restrictionmeans that the external conditions represented by the point of the base b ∈ B willremain constant under the motion. Example 2.1.
Consider a system represented by n pairwise distinct points in the k -dimensional space z , . . . , z n ∈ R k moving in the complement of a set of obstacles o , o , . . . , o m ∈ R k which are also pairwise distinct. In practical situations one has k = 3 or k = 2. Hence, z i = z j , o i = o j (for i = j ), and z i = o j . In the formalism ofparametrised motion planning, the base space B = F ( R k , m ) is the configurationspace of m distinct ordered points in R k representing the obstacles, the total space E = F ( R k , n + m ) is the configuration space of tuples ( z , . . . , z n , o , . . . , o m ) andthe projection p : E → B is given by p ( z , . . . , z n , o , . . . , o m ) = ( o , . . . , o m ). Amotion planning algorithm produces for two configurations, ( z , . . . , z n , o , . . . , o m )and ( z ′ , . . . , z ′ n , o , . . . , o m ), a continuous motion( z ( t ) , . . . , z n ( t ) , o , . . . , o m ) ∈ F ( R k , n + m ) , where t ∈ [0 , , with z i (0) = z i and z i (1) = z ′ i for i = 1 , . . . , n . Observe that the positions of theobstacles are not assumed to be known in advance but are rather part of the inputof the problem to be solved by the planner.For a fibration p : E → B with fibre X , let E IB denote the space of all continuouspaths γ : I → E lying in a single fibre of p , so that the path p ◦ γ is constant. Denoteby E × B E = { ( e, e ′ ) ∈ E × E ′ | p ( e ) = p ( e ′ ) } the space of pairs of configurationslying in the same fibre. The map Π : E IB → E × B E , γ ( γ (0) , γ (1)), is a fibration,with fibre Ω X , the space of based loops in X . Recall that we assume the fibre X to be path-connected. We define the parameterised topological complexity of thefibration p : E → B to be the sectional category of the associated fibration Π, i.e. TC [ p : E → B ] := secat (Π) . If the fibration p is clear from the context, we will sometimes use the abbreviatednotation TC [ p : E → B ] = TC B ( X ) emphasising the role of the fibre X .We shall see below that the parametrised topological complexity of a fibration p : E → B with fibre X can be strictly greater than the topological complexity TC ( X ), in particular this is the case in the situation of Example 2.1 - collision freemotion planning with multiple obstacles. For a topological space Y , let F ( Y, n ) = { ( x , . . . , x n ) ∈ Y n | x i = x j if i = j } be the configuration space of n distinct ordered points in Y . As is well known,if Y is a manifold without boundary of dimension at least two, the forgetful map p : F ( Y, n + m ) → F ( Y, m ), p ( x , . . . , x n + m ) = ( x , . . . , x m ), is a fibration, the OPOLOGY OF PARAMETRISED MOTION PLANNING ALGORITHMS 5
Fadell-Neuwirth bundle, with fibre F ( Y r { m points } , n ). Results of Section 9below yield that for k = 3 or more generally for any odd k ≥ TC [ p : F ( R k , n + m ) → F ( R k , m )] = 2 n + m − TC ( F ( R k r { m points } , n )) = 2 n of the fibre found in [9]. The difference between the parametrised and non-parametrised topological complexities TC B ( X ) − TC ( X ) in this example equals m −
1; hence this difference may be arbitrarily large.3.
Sectional category and its generalised version
In this section, we review notions of sectional category of a fibration. Theconcept of sectional category was originally introduced by A. Schwarz [17], whoused the term “genus”of a fibration.Let p : E → B be a fibration ; this means that p is a continuous map satisfyingthe homotopy lifting property with respect to any space, [18]. Definition 3.1.
The sectional category of p is the smallest integer k ≥ B admits an open cover B = U ∪ U ∪ · · · ∪ U k with the property thateach set U i admits a continuous section s i : U i → E ; here s i is a continuous mapwith p ◦ s i : U i → B equal to the inclusion id U i : U i → B .We shall denote the sectional category of p by secat ( p ) or more informatively secat ( p : E → B ). Note that secat ( p : E → B ) = 0 if and only if p admits a globallydefined continuous section.If the base B is locally contractible (which is a typical situation for this article)then any point b ∈ B has a neighbourhood U ⊂ B with a continuous section U → E and the sectional category secat ( p ) depends only on the global topologicalstructure of p .There is also a notion of generalised sectional category of a fibration p : E → B ,which we denote by secat g ( p : E → B ) and abbreviate secat g ( p ): Definition 3.2.
The generalised sectional category secat g ( p : E → B ) of thefibration p : E → B is defined as the smallest integer k ≥ B admits a partition into k + 1 subsets B = A ⊔ A ⊔ · · · ⊔ A k , A i ∩ A j = ∅ for i = j, with the property that each A i admits a continuous section s i : A i → E . Notethat here we impose no restrictions on the nature of the sets A i ; in particular wedo not require these sets to be open. D. COHEN, M. FARBER, AND S. WEINBERGER
Clearly one has secat ( p ) ≥ secat g ( p ). In a recent paper, Garc´ıa-Calcines [11](see also [19]) proved that for a fibration p : E → B with both spaces E and B being ANRs one has in fact the equality secat ( p : E → B ) = secat g ( p : E → B ) . (3.1)Recall that a metrisable topological space X is an absolute neighborhood retract (ANR) if for any homeomorphism h : X → Y mapping X onto a closed subset h ( X ) of a metrisable topological space Y there exists an open neighbourhood U of h ( X ) ⊂ Y which retracts onto h ( X ). Refer to Borsuk [1] for a detailed discussion.Well known facts concerning ANRs include:(i) Any ANR is locally contractible;(ii) Any polyhedron is an ANR;(iii) A metrisable topological space is an ANR if it can be represented as theunion of countably many open subsets which are ANRs.For applications in robotics one may always restrict attention to the class ofANR spaces.For convenience of the reader we recall the well-known cohomological lowerbound for the sectional category: Proposition 3.3.
Let p : E → B be a fibration and let R be a ring. Supposethat a set of cohomology classes u , u , . . . , u k ∈ H ∗ ( B, R ) satisfying p ∗ u i = 0 for i = 0 , , . . . , k has nontrivial product u ⌣ u ⌣ · · · ⌣ u k = 0 ∈ H ∗ ( B, R ) . Then secat ( p : E → B ) is greater than k .Proof. Assuming the contrary, secat ( p ) ≤ k , let B = U ∪ U ∪ · · · ∪ U k be an opencover with the property that each U i admits a homotopy section s i : U i → E .Then u i | U i = s ∗ i p ∗ ( u i ) = 0 and hence u i can be lifted to a relative cohomologyclass ˜ u i ∈ H ∗ ( B, U i , R ), i.e. u i = ˜ u i | B . The product ˜ u ⌣ ˜ u ⌣ · · · ⌣ ˜ u k lies inthe trivial group H ∗ ( B, ∪ ki =0 U i , R ) = H ∗ ( B, B, R ) = 0 and hence the product u ⌣ u ⌣ · · · ⌣ u k = (˜ u ⌣ ˜ u ⌣ · · · ⌣ ˜ u k ) | B = 0vanishes which contradicts our assumptions. (cid:3) The concept of pararmeterised topological complexity
In this section, we give the general definition of parameterised topological com-plexity, and discuss some initial examples and results.Let p : E → B be a fibration. We wish to define a topological invariant measur-ing the complexity of motion planning algorithms in the family of configurationspaces parameterised by p . Recall that such an algorithm is a rule taking pairs of OPOLOGY OF PARAMETRISED MOTION PLANNING ALGORITHMS 7 points e , e ∈ E with p ( e ) = p ( e ) = b ∈ B to a continuous path γ : I → X b inthe fibre X b = p − ( b ) with γ (0) = e and γ (1) = e .Denote by E IB the space of all continuous paths γ : I → E such that the path p ◦ γ is constant; these are paths lying in single fibre of E . Let E × B E ⊂ E × E bethe space of pairs of configurations ( e, e ′ ) lying in the same fibre, i.e., E × B E = { ( e, e ′ ) ∈ E × E ′ | p ( e ) = p ( e ′ ) } . Sending a path to its endpoints defines a mapΠ : E IB → E × B E, γ ( γ (0) , γ (1)) , (4.1)Clearly, Π is a fibration with fibre Ω X , the space of based loops in X , where X isthe fibre of p : E → B . Definition 4.1.
The parameterised topological complexity TC [ p : E → B ] of thefibration p : E → B is defined as the sectional category of Π : E IB → E × B E , TC [ p : E → B ] := secat (Π : E IB → E × B E ) . Explicitly, TC [ p : E → B ] is equal to the smallest integer k ≥ E × B E admits an open cover(4.2) E × B E = U ∪ U ∪ · · · ∪ U k , and the map Π : E IB → E × B E admits a continuous section s i : U i → E IB for each i = 0 , , . . . , k .We occasionally use the more compact notation TC B ( X ) for the parameterisedtopological complexity TC B ( X ) = TC [ p : E → B ] = secat (Π : E IB → E × B E ) . For a fibration p : E → B and a subset B ′ ⊂ B , let E ′ = p − ( B ′ ), and let p ′ : E ′ → B ′ denote the restricted fibration over B ′ . Then we obviously have(4.3) TC [ p : E → B ] ≥ TC [ p ′ : E ′ → B ′ ] , or in abbreviated notation, TC B ( X ) ≥ TC B ′ ( X )for B ′ ⊂ B . In particular, if B ′ = { b } is a single point, we obtain(4.4) TC B ( X ) ≥ TC ( X ) . In sections Sections 8, 9 of this paper we shall see examples where strict inequalityholds in (4.4). Moreover, we shall see that the difference TC B ( X ) − TC ( X ) can bearbitrarily large.Next we consider a few examples where (4.4) is satisfied as an equality. D. COHEN, M. FARBER, AND S. WEINBERGER
Example 4.2.
Suppose that E = X × B it the trivial fibration. In engineeringterms it means that the external conditions are invariable (not changing). Then E × B E = X × X × B and E IB = X I × B . In this case fibration (4.1) reduces to p × id : X I × B → X × X × B where p : X I → X × X is given by p ( γ ) = ( γ (0) , γ (1)).Clearly the Schwarz genus of p × id is equal to TC ( X ). We see that in this case TC B ( X ) = TC ( X ), i.e., trivial parametrisation does not add complexity. Proposition 4.3.
Let p : E → B be a principal bundle with group G (a connectedtopological group). Then TC B ( G ) = cat ( G ) = TC ( G ) . Proof.
Consider the homeomorphisms F : E × G → E × B E, F ( e, g ) = ( e, ge ) , and F ′ : E × P ( G ) → E IB , F ′ ( e, γ ) = ( t γ ( t ) e ) . Here P ( G ) denotes the spaces of continuous paths γ : I → G with γ (0) = 1 ∈ G .Let p : P ( G ) → G be p ( γ ) = γ (1). From the commutative diagram E × P ( G ) F ′ / / id × p (cid:15) (cid:15) E IB Π (cid:15) (cid:15) E × G F / / E × B E the sectional category of Π is equal to the sectional category of id × p . The latterclearly coincides with the sectional category of p which is cat ( G ), according tothe definition. (cid:3) Example 4.4.
Consider the Hopf fibration S → S with fibre X = S . In thiscase B = S and TC ( X ) = TC ( S ) = 1. By Proposition 4.3, TC B ( S ) = 1 and weshall describe a specific motion planner. We think of S as being the set of pairs( z , z ) ∈ C × C satisfying | z | + | z | = 1. The group S = { u ∈ C | | u | = 1 } acts on S by u · ( z , z ) = ( uz , uz ) and the base S is the space of orbits ofthis action. We may represent the base S as C ∪ {∞} and then the projection p : S → S is given by p ( z , z ) = z · z − .We shall describe a parametrised motion planning algorithm with two open sets S × S S = U ∪ U . Here U will denote the set of all pairs (( z , z ) , ( z ′ , z ′ )) satisfying z z = z ′ z ′ and( z , z ) = ( − z , − z ). The set U consists of all pairs (( z , z ) , ( z ′ , z ′ )) satisfying OPOLOGY OF PARAMETRISED MOTION PLANNING ALGORITHMS 9 z z = z ′ z ′ and ( z , z ) = ( z , z ). The section s : U → ( S ) IS can be defined by s (( z , z ) , ( z ′ , z ′ )) = ( t ( e itϕ z , e itϕ z ))where ϕ ∈ ( − π, π ) satisfies z ′ = e iϕ z and z ′ = e iϕ z . The section s : U → ( S ) IS can be defined by the formula s (( z , z ) , ( z ′ , z ′ )) = ( t ( e itϕ z , e itϕ z ))where ϕ ∈ (0 , π ) satisfies z ′ = e iϕ z and z ′ = e iϕ z . Proposition 4.5. If p : E → B is a fibration with path-connected fibre X , and TC B ( X ) = 0 , then X is contractible. Conversely, if the fibre X is contractible and E × B E is homotopy equivalent to a CW-complex then TC B ( X ) = 0 .Proof. The first statement follows from (4.4) and the known fact that TC ( X ) = 0 ifand only if X is contractible. The second statement follows by applying obstructiontheory. The fibre of fibration (4.1) is the loop space Ω X , which is contractible if X is contractible. (cid:3) Remark . The case of Proposition 4.5 when the fibres of p : E → B are convexsets is trivial, however a slightly more general situation when the fibres are star-likeis already not obvious.Our main Definition 4.1 defines parametrised topological complexity using opencovers of E × B E . We note that: Proposition 4.7. If p : E → B is a locally trivial fibration and the spaces E and B are metrisable separable ANRs then in Definition 4.1 instead of open covers onemay use arbitrary covers of E × B E or, equivalently, arbitrary partitions E × B E = F ⊔ F ⊔ · · · ⊔ F k , F i ∩ F j = ∅ admitting continuous sections s i : F i → E IB where i = 0 , , . . . , k . In other words, TC [ p : E → B ] = secat g (Π : E IB → B × B E ) . Proof.
Due to the result (3.1) of Garc´ıa-Calcines, Proposition 4.7 follows once wehave shown that under our assumptions the spaces E IB and E × B E are ANRs.We note that the fibre X of p : E → B is an ANR, being a neighbourhoodretract of E . Secondly, X × X is an ANR, by [1, Chapter IV, Theorem (7.2)].And finally, applying [1, Chapter IV, Theorem (10.5)], E × B E is an ANR as itis the total space of a locally trivial fibration E × B E → B with fibre X × X obtained by pulling back the product fibration E × E → B × B along the diagonal∆ B : B → B × B . Next, we note that the map E IB → B is a locally trivial fibration with fibre X I .Indeed, if U ⊂ B is an open subset such that p : E → B is trivial over U thenthe fibration E IB → B is also trivial over U . Thus the space E IB is the total spaceof a locally trivial fibration with base B and fibre X I . We observe that X I is anANR by [1, Chapter IV, Theorem (5.1)]. Now, since we know that both spaces B and X I are ANRs we obtain that E IB is ANR as well, by [1, Chapter IV, Theorem(10.5)]. This completes the proof. (cid:3) Homotopy invariance
Proposition 5.1.
If fibrations p : E → B and p ′ : E ′ → B are fibrewise homotopyequivalent then TC [ p : E → B ] = TC [ p ′ : E ′ → B ] . Proposition 5.1 follows from the following observation:
Proposition 5.2.
Consider the commutative diagram E ′ p ′ ❆❆❆❆❆❆❆❆ f / / E p (cid:127) (cid:127) ⑦⑦⑦⑦⑦⑦⑦⑦ g o o B where p, p ′ are fibrations, p ′ = pf , p = p ′ g and g ◦ f : E ′ → E ′ is fibrewise homotopicto the identity map E ′ : E ′ → E ′ . Then TC [ p ′ : E ′ → B ] ≤ TC [ p : E → B ] .Proof. Let h t : E ′ → E ′ be a homotopy satisfying p ′ ◦ h t = p ′ and h = 1 E ′ while h = g ◦ f . Let U ⊂ E × B E be a subset with a continuous section s : U → E IB of Π. Consider the set V = ( f × f ) − ( U ) ⊂ E ′ × B E ′ . We want to describe acontinuous section s ′ : V → ( E ′ ) IB of the fibration Π ′ : ( E ′ ) IB → E ′ × B E ′ . For apair ( a, b ) ∈ V and τ ∈ [0 ,
1] we set s ′ ( a, b )( τ ) = h τ ( a ) , for 0 ≤ τ ≤ / ,g ( s ( f ( a ) , f ( b ))(3 τ − , for 1 / ≤ τ ≤ / ,h − τ ) ( b ) , for 2 / ≤ τ ≤ . This proves our statement. (cid:3) Product Inequality
In this section we establish a product inequality for parameterised topologicalcomplexity.
OPOLOGY OF PARAMETRISED MOTION PLANNING ALGORITHMS 11
Proposition 6.1.
Let p ′ : E ′ → B ′ and p ′′ : E ′′ → B ′′ be fibrations with path-connected fibres X ′ and X ′′ respectively. Consider the product fibration p : E → B ,with fibre X , where B = B ′ × B ′′ , E = E ′ × E ′′ , X = X ′ × X ′′ and p = p ′ × p ′′ .Then, TC [ p : E → B ] ≤ TC [ p ′ : E ′ → B ′ ] + TC [ p ′′ : E ′′ → B ′′ ] . Equivalently, in abbreviated notation, TC B ′ × B ′′ ( X ′ × X ′′ ) ≤ TC B ′ ( X ′ ) + TC B ′′ ( X ′′ ) . Proof.
Let Π : E IB → E × B E , Π ′ : ( E ′ ) IB ′ → E ′ × B ′ E ′ , Π ′′ : ( E ′′ ) IB ′′ → E ′′ × B ′′ E ′′ denote the fibrations of (4.1). It is readily checked that E IB = ( E ′ ) IB ′ × ( E ′′ ) IB ′′ , E × B E = ( E ′ × B ′ E ′ ) × ( E ′′ × B ′′ E ′′ ), and that the fibration Π : E IB → E × B E isequivalent to the product fibrationΠ ′ × Π ′′ : ( E ′ ) IB ′ × ( E ′′ ) IB ′′ → ( E ′ × B ′ E ′ ) × ( E ′′ × B ′′ E ′′ ) . Since the sectional category of the product fibration is at most the sum of thesectional categories of the constituent fibrations (see [17, Proposition 22]), theresult follows. (cid:3)
Corollary 6.2.
Let p ′ : E ′ → B and p ′′ : E ′′ → B be fibrations over base B with fibres X ′ and X ′′ respectively. Let p : E → B be the fibration with fibre X = X ′ × X ′′ , where E = E ′ × B E ′′ and p = p ′ × B p ′′ . Then, TC [ p : E → B ] ≤ TC [ p ′ : E ′ → B ] + TC [ p ′′ : E ′′ → B ] . Equivalently, in abbreviated notation, TC B ( X ′ × X ′′ ) ≤ TC B ( X ′ ) + TC B ( X ′′ ) . Proof.
Identify B with its image under the diagonal ∆ : B → B × B in the base ofthe fibration p ′ × p ′′ : E ′ × E ′′ → B × B . The result then follows from Proposition6.1 by applying inequality (4.3). (cid:3) Upper and lower bounds
Let X denote the fibre of a fibration p : E → B . We note the following obviousinequality: TC B ( X ) ≤ cat ( E × B E ) . (7.1)Let dim( Y ) denote the covering dimension of a topological space Y . We shall alsouse the notation hdim ( Y ) for the minimum dimension dim Z where Z is a spacehomotopy equivalent to Y . We shall refer to hdim ( Y ) as to homotopical dimension of Y . Using the well-known properties of the Lusternik-Schnirelmann category wecan write TC B ( X ) ≤ hdim ( E × B E ) , (7.2)as follows from (7.1). Proposition 7.1.
Let p : E → B be a locally trivial fibration with path-connectedfibre X . Assume that the topological spaces E and B are metrisable. Then one has TC B ( X ) = TC [ p : E → B ] ≤ X ) + dim( B ) . Proof.
We apply the general upper bound for the sectional category [17] in termsof the dimension of the base, TC B ( X ) ≤ dim( E × B E ). Next we note that E × B E is the total space of a locally trivial fibration over B with fibre X × X implyingdim( E × B E ) ≤ dim( X × X ) + dim B ≤ X + dim B , cf. [3, Chapter 7]. (cid:3) The upper bound of Proposition 7.1 can be improved under connectivity as-sumptions on the fibre X of p : E → B : Proposition 7.2.
Let p : E → B be a locally trivial fibration with fibre X wherethe spaces E, B, X are CW-complexes. Assume that the fibre X is r -connected.Then TC B ( X ) < hdim ( E × B E ) + 1 r + 1 ≤ X + dim B + 1 r + 1 . Proof.
The fibre of the fibration Π : E IB → E × B E (see (4.1)) is Ω X ; it is ( r − (cid:3) Proposition 7.3.
Let p : E → B be a fibration with path-connected fibre. Considerthe diagonal map ∆ : E → E × B E , where ∆( e ) = ( e, e ) . Then the parameterisedtopological complexity TC [ p : E → B ] = TC B ( X ) is greater than or equal to thecup length of the kernel ker[∆ ∗ : H ∗ ( E × B E ; R ) → H ∗ ( E ; ∆ ∗ R )] where R is anarbitrary system of coefficients on E × B E . In other words, if for some cohomologyclasses u , . . . , u k ∈ H ∗ ( E × B E ; R ) satisfying ∆ ∗ ( u i ) = 0 , i = 1 , . . . , k the cup-product u ⌣ u ⌣ · · · ⌣ u k = 0 is nonzero then TC B ( X ) ≥ k . OPOLOGY OF PARAMETRISED MOTION PLANNING ALGORITHMS 13
Proof.
Consider the commutative diagram E ∆ ❋❋❋❋❋❋❋❋❋❋ c / / E IB Π { { ✈✈✈✈✈✈✈✈✈ E × B E where the map c : E → E IB associates with each e ∈ E the constant path c ( e )( t ) = e . Clearly c is a homotopy equivalence, and Π ◦ c = ∆. Therefore, the kernel ofΠ ∗ : H ∗ ( E × B E ; R ) → H ∗ ( E IB ; Π ∗ R ) coincides withker[∆ ∗ : H ∗ ( E × B E ; R ) → H ∗ ( E ; ∆ ∗ R )] . (7.3)Our claim now follows from Proposition 3.3. (cid:3) Our next goal is to relate the kernel (7.3) with the ideal of the zero-divisors ofthe fibre X , i.e. withker[ ⌣ : H ∗ ( X ; R ) ⊗ H ∗ ( X ; R ) → H ∗ ( X, R )] . (7.4)Recall the classical Leray-Hirsch theorem [14, Theorem 4D.1]. Let p : E → B bea locally trivial fibration with fibre X and let R be a commutative coefficient ring(typically Z or a field). Assume that:(a) For each q ≥
0, the cohomology H q ( X ; R ) is a free finitely generated R -module.(b) There exist cohomology classes c j ∈ H ∗ ( E ; R ) such that their restrictions ι ∗ ( c j ) form a free basis of H ∗ ( X ; R ) for each fibre X , where ι : X → E denotes the inclusion.The Leray-Hirsch theorem states that under these hypotheses the cohomology ofthe total space H ∗ ( E ; R ) is a free H ∗ ( B ; R )-module with basis { c j } . Explicitly,the map(7.5) Φ : H ∗ ( B ; R ) ⊗ R H ∗ ( X ; R ) → H ∗ ( E ; R ) , X b i ⊗ ι ∗ ( c j ) X p ∗ ( b i ) ⌣ c j , is an isomorphism. Proposition 7.4.
If a locally trivial fibration p : E → B with fibre X satisfiesthe Leray-Hirsch theorem and the cohomology of the base H ∗ ( B ; R ) is flat as an R -module then the kernel ker[∆ ∗ : H ∗ ( E × B E ; R ) → H ∗ ( E ; ∆ ∗ R )] is isomorphic,as an H ∗ ( B ) -module, to H ∗ ( B ) ⊗ R ker[ ⌣ : H ∗ ( X ; R ) ⊗ H ∗ ( X ; R ) → H ∗ ( X, R )] . (7.6) Proof.
Applying the K¨unneth formula, we see that the cohomology classes C j,j ′ = f ∗ ( c j × c j ′ ) ∈ H ∗ ( E × B E ; R )restrict to a free basis on each fibre X × X of the fibration p ′ : E × B E → B where f : E × B E → E × E denotes the inclusion; therefore the fibration p ′ satisfies theassumptions of the Leray-Hirsch theorem as well. We obtain an isomorphismΦ ′ : H ∗ ( B ; R ) ⊗ H ∗ ( X × X ; R ) → H ∗ ( E × B E ; R )which appears in the following commutative diagram H ∗ ( B ; R ) ⊗ R H ∗ ( X × X ; R ) Φ ′ ≃ / / ⊗ ∆ ∗ X (cid:15) (cid:15) H ∗ ( E × B E ; R ) ∆ ∗ (cid:15) (cid:15) H ∗ ( B ; R ) ⊗ R H ∗ ( X ; R ) Φ ≃ / / H ∗ ( E × B E ; R ) . Here ∆ X : X → X × X is the diagonal. Since H ∗ ( B ; R ) is flat as an R-module weobtain that the kernel of ∆ ∗ coincides (after applying the isomorphisms Φ ′ and Φ)with ker(1 ⊗ ∆ ∗ X ) = 1 × ker( ⌣ ). (cid:3) This result may seem to suggest that under the assumptions of the Leray-Hirschtheorem the cup-length of the kernel (7.3) equals the zero-divisors cup-length of thefibre. We shall see below that it is not the case. The isomorphism of Proposition7.4 is only additive and in reality the multiplicative structure of ker(∆ ∗ ) is adeformation of the ideal of the zero-divisors of the fibre.8. Motion planing for a robot in 3-space and two obstacles withunknown in advance positions
Here we consider the situation of parametrised motion planning in 3-dimensionalspace with two floating obstacles, it is a special case of the situation considered inExample 2.1. This discussion is intended to illustrate our general Theorem 9.1.The obstacles are represented by two distinct points o , o ∈ R k where k = 3.The position of the robot is represented by z ∈ R k −{ o , o } . The relevant fibrationis the Fadell-Neuwith fibration(8.1) p : E = F ( R k , → B = F ( R k , , p ( o , o , z ) = ( o , o ) . The fibre of this locally trivial fibration has the homotopy type of a bouquet oftwo spheres X = p − ( o , o ) ∼ = { z ∈ R k | z = o , z = o } ≃ S k − ∨ S k − . OPOLOGY OF PARAMETRISED MOTION PLANNING ALGORITHMS 15
The base space B = F ( R k , ≃ S k − , has the homotopy type of a sphere, andthe total space E = F ( R k ,
3) has the homotopy type of a 2( k − X ≃ S k − ∨ S k − of the fibration (8.1) is ( k − k ≥
2, the parameterised topological complexityadmits the upper bound(8.2) TC [ p : F ( R k , → F ( R k , ≤ . We show below that for k = 3 one has TC [ p : F ( R k , → F ( R k , . Note thatthe (unparameterised) topological complexity of the fibre is TC ( X ) = 2, see [7].Below we consider integral cohomology, skipping Z in the notation. It is well-known [5] that the space F ( R k ,
3) has three ( k − ω , ω , ω which satisfy the following relations ω ij = 0 , and ω ω = ω ( ω − ω )(8.3)and generate the cohomology ring. The class ω ij is defined as the pull-back of thefundamental class u ∈ H k − ( F ( R k , i -th and j -thparticles of the configuration; here the index i = 1 corresponds to o , the index i = 2 corresponds to o and the index i = 3 corresponds to z . The classes ω and ω restrict to a set of free generators of the fibre; hence we see that fibration (8.1)satisfies the assumptions of the Leray-Hirsch theorem. The class ω generates thecohomology of the base B . By the Leray-Hirsch theorem the following classes forman additive base of the integral cohomology of F ( R k ,
3) in positive degrees: ω , ω , ω , ω ω , ω ω . The first 3 classes have degree ( k −
1) and the last 2 classes have degree 2( k − ω ω through thegenerators in degree 2( k −
1) mentioned above. Note that the products ω ω and ω ω vanish when restricted to fibre.The three term relation (8.3) represents the cup-product of the total space H ∗ ( E ) as a deformation of the tensor product algebra H ∗ ( B ) ⊗ H ∗ ( X ).Next we consider cohomology of the space E × B E which can be identified withthe configuration space of tuples ( o , o , z, z ′ ) ∈ ( R k ) satisfying o = o , z = o , z = o , z ′ = o , z ′ = o . By Proposition 7.4 the fibration p ′ : E × B E → B satisfiesthe assumptions of the Leray-Hirsch theorem and hence the additive structure ofthe cohomology is given by 5 classes ω , ω , ω , ω ′ , ω ′ (8.4) of degree ( k − k − ω ω ′ , ω ω ′ , ω ω ′ , ω ω ′ , ω ω , ω ω , ω ω ′ , ω ω ′ , (8.5)and by 4 classes of degree 3( k − ω ω ω ′ , ω ω ω ′ , ω ω ω ′ , ω ω ω ′ . (8.6)Multiplicatively these classes satisfy the following relations ω ij = 0 , ω ′ ij = 0and two three term relations ω ω = ω ( ω − ω ) , ω ′ ω ′ = ω ( ω ′ − ω ′ ) . (8.7)The first relation (8.7) follows from the fact that the classes ω ij are pull-backs ofthe corresponding classes in H ∗ ( E ) under the projection on the first coordinate;similarly for the second relation in (8.7).Finally we examine the map ∆ : E → E × B E and the homomorphism inducedon cohomology. Since ∆ ∗ ( ω ij ) = ∆ ∗ ( ω ′ ij ) we see that the classes ω − ω ′ and ω − ω ′ lie in the kernel of ∆ ∗ . We claim that the product( ω − ω ′ ) ( ω − ω ′ ) ∈ H ∗ ( E × B E )(8.8)is nonzero. Note that here we use our assumption that k = 3 since for k even thesquare above would vanish. Our claim is equivalent to the non-vanishing of theproduct ω ω ′ ( ω − ω ′ ) = ω ω ′ ω − ω ω ′ ω ′ = ω ω ω ′ − ω ′ ω ′ ω = ω ( ω − ω ) ω ′ − ω ( ω ′ − ω ′ ) ω = ω ω ω ′ − ω ω ω ′ . We see that this class is the difference of two distinct generators from the list(8.6) and hence it is nonzero. Applying Proposition 7.3 combined with the upperbound (8.2) we obtain TC [ p : F ( R k , → F ( R k , k = 3. Note that thisargument applies without modifications to any odd dimension k ≥ the input ofthe algorithm. OPOLOGY OF PARAMETRISED MOTION PLANNING ALGORITHMS 17 Obstacle-avoiding collision-free motion of multiple robots in thepresence of multiple obstacles with unknown in advance positions
In this section we generalise the result of the previous section in several direc-tions: firstly we allow multiple robots moving collision free, secondly we allow anarbitrary number of obstacles. Our main focus is on the case of 3-dimensionalunderlying space however our results are applicable more generally to Euclideanspace R k of any dimension k with the only restriction that k must be odd.In the case when the dimension k is even the final answer is slightly different,it requires different lower and upper bounds and will be described in a separatepublication.Our setting is as follows: we consider m ≥ R k represented bypairwise distinct points z , . . . , z m , there are also n robots represented by the points z m +1 , . . . , z m + n ∈ R k , these points must be pairwise distinct and distinct from thepositions of the obstacles. As described in Example 2.1 we have to consider theFadell-Neuwirth fibration p : F ( R k , n + m ) → F ( R k , m ) , (9.1)where ( z , . . . , z m , z m +1 , . . . , z m + n ) ( z , . . . , z m ) and compute its parametrisedtopological complexity. In the previous section we considered the special case m = 2, n = 1 and k = 3. It will be convenient to use the notation p : E → B for(9.1); the fibre F ( R k − O m , n ) of this fibration will be denoted by X . Here O m denotes a configuration of m distinct points representing the obstacles.To explain our assumption m ≥ m = 1 the baseof the fibration (9.1) is contractible and hence the fibration is trivial. Example4.2 shows that in this case the parametrised topological complexity coincides with TC ( F ( R k − O , n )) which is known [9].In this section we shall prove the following theorem which can be viewed themain result of this paper. Theorem 9.1.
Let k ≥ be odd. The parameterised topological complexity of themotion of n ≥ non-colliding robots in R k in the presence of m ≥ non-collidingobstacles is equal to n + m − . In other words, the parameterized topologicalcomplexity of the Fadell-Neuwirth bundle p : F ( R k , n + m ) → F ( R k , m ) is (9.2) TC (cid:2) p : F ( R k , n + m ) → F ( R k , m ) (cid:3) = 2 n + m − . Note that the standard (i.e. nonparametrised) topological complexity of thefibre of the Fadell-Neuwirth fibration is 2 n , see Theorems 5.1 and 6.1 of [9]. Thuswe see that the parametrised topological complexity exceeds the standard one byapproximately the number of obstacles. The rest of this section is devoted to the proof of Theorem 9.1.First, we apply Proposition 7.2 to get an upper bound. We note that the fibre X of the Fadell-Neuwirth fibration (9.1) is ( k − hdim X = ( k − n ,while the homotopical dimension of the base is hdim B = ( k − m − E × B E has homotopical dimension ( k − n + m − TC (cid:2) p : E → B (cid:3) < n + m − k − , that is(9.3) TC (cid:2) p : F ( R k , n + m ) → F ( R k , m ) (cid:3) ≤ n + m − . This gives an upper bound in (9.2).The proof of Theorem 9.1 will use the structure of the integral cohomology ring H ∗ ( E × B E ), which we describe next. Proposition 9.2.
The integral cohomology ring H ∗ ( E × B E ) contains degree k − classes ω ij , ω ′ ij where ≤ i < j ≤ n + m, which satisfy the following relations ω ij = ( ω ′ ij ) = 0 for all i < j,ω ik ω jk = ω ij ( ω jk − ω ik ) for all i < j < k,ω ′ ik ω ′ jk = ω ′ ij ( ω ′ jk − ω ′ ik ) for all i < j < k,ω ij = ω ′ ij for all ≤ i < j ≤ m. Proof.
A point of the space E × B E can be represented by an ( m + 2 n )-tuple ofthe form ( z , . . . , z m , z m +1 , . . . , z m + n , z ′ m +1 , . . . , z ′ m + n ) ∈ ( R k ) m +2 n . where the first m points represent m obstacles, and the tuples ( z m +1 , . . . , z m + n )and ( z ′ m +1 , . . . , z ′ m + n ) represent the initial and the final configurations of therobots. Clearly, we require that for any i < jz i = z j and z ′ i = z ′ j , (9.4)and besides z i = z ′ j for i ≤ m ;(9.5)note that index j in (9.5) automatically satisfies j > m . Conditions (9.4) and (9.5)guarantee that no collisions between robots and between obstacles happen. OPOLOGY OF PARAMETRISED MOTION PLANNING ALGORITHMS 19
For any pair of indexes i < j satisfying either (9.4) or (9.5) we may view thecorresponding pair of points as an element of the configuration space F ( R k ,
2) ofpairs of distinct points in R k . It is well-known that F ( R k ,
2) is homotopy equivalentto the sphere S k − and has therefore a fundamental class in H k − ( F ( R k , i < j one defines the class ω ij ∈ H k − ( E × B E ) as the pull-back ofthe fundamental class. Similarly, for i ≤ m and j > m one defines the class ω ′ ij ∈ H k − ( E × B E ) as the pull-back of the fundamental class under the map E × B E → F ( R k ,
2) projecting E × B E onto the pairs ( z i , z ′ j ). Besides, for m
2) onthe pair ( z ′ i , z ′ j ). Finally we formally define ω ′ ij for i < j ≤ m by setting ω ′ ij = ω ij .All the relations mentioned in the statement of Proposition 9.2 are well knownto hold in cohomology of configuration spaces, see [5]. Since our classes are pull-backs of cohomology classes originating from configuration spaces these relationshold as well. (cid:3) Next we introduce the following notations. We shall consider sequences I =( i , i , . . . , i p ) and J = ( j , j , . . . , j p ) consisting of elements of the index set { , , . . . , m + n } such that i s < j s for all s = 1 , . . . , p ; we shall express this by I < J for brevity.For each such pair
I < J we define a cohomology class ω IJ = ω i j ω i j . . . ω i p j p ∈ H ( k − p ( E × B E )as the cup-product of the classes ω ij defined above. The classes ω ′ IJ ∈ H ( k − p ( E × B E )are defined similarly where instead of the classes ω ij one takes the classes ω ′ ij . Thecase p = 0 is also allowed; in this case we define ω IJ = 1 = ω ′ IJ .A sequence J = ( j , j , . . . , j p ) is said to be increasing if j < j < · · · < j p . Proposition 9.3.
A free basis of the abelian group H ∗ ( E × B E ) is given by theset of cohomology classes of the form ω I J ω I J ω ′ I J , (9.6) where I < J , I < J , I < J , the sequences J , J , J are increasing and J takes values in { , . . . , m } , while J and J take values in { m + 1 , . . . , m + n } .Proof. We want to apply the Leray-Hirsch theorem to the fibration E × B E → B .Recall that E = F ( R k , n + m ) and B = F ( R k , m ) and the fibre of this fibrationis X × X where X = F ( R k − O m , n ). The classes ω ij with i < j ≤ m originate from the base of this fibration. Moreover, it is well known that the cohomology ofthe base H ∗ ( B ) = H ∗ ( F ( R k , m )) has a free additive basis consisting of the classes ω IJ where I < J run over all sequences of elements of the set { , , . . . , m } suchthat the sequence J = ( j , j , . . . , j p ) is increasing . Clearly here p must be at most m − ω I J and ω ′ I J where I < J , I < J and theincreasing sequences J and J consist of elements of the set { m + 1 , . . . , m + n } .Using the known results about the cohomology algebras of configuration spaces(see [5], Chapter V, Theorems 4.2 and 4.3) as well as the K¨unneth theorem we seethat the restrictions of the family of classes ω I J ω ′ I J onto the fibre form a freebasis in the cohomology of the fibre H ∗ ( X × X ). Hence, applying the Leray-Hirschtheorem [14], we obtain that a free basis of the cohomology H ∗ ( E × B E ) is givenby the set of classes of the form ω I J ω I J ω ′ I J (9.7)where I < J , I < J , I < J , the sequences J , J , J are increasing and J takes values in { , . . . , m } while J and J take values in { m + 1 , . . . , m + n } . (cid:3) Using Proposition 9.3 one may show that the classes ω ij , ω ′ ij are multiplicativegenerators of the cohomology ring H ∗ ( E × B E ) and the relations mentioned inProposition 9.2 form a defining set of relation. However we shall not need thisstatement in this paper.Note that the maximal degree of the class (9.7) is 2 n + m − J = (2 , , . . . , m ) and J = J = ( m + 1 , . . . , m + n ).Next we consider the diagonal map ∆ : E → E × B E . Proposition 9.4.
For any i < j the class ω ij − ω ′ ij (9.8) lies in the kernel of the homomorphism ∆ ∗ : H ∗ ( E × B E ) → H ∗ ( E ) induced bythe diagonal map ∆ : E → E × B E .Proof. The points of the space E = F ( R k , n + m ) are represented by configurations( z , z , . . . , z n + m ) of pairwise distinct points of R k and ∆ maps such a configura-tion to ( z , z , . . . , z m , z m +1 , . . . , z n + m , z m +1 , . . . , z n + m ) (where the last n pointsare repeated twice. Our statement now follows from the explicit description of theclasses ω ij and ω ′ ij given in the proof of Proposition 9.2. (cid:3) Note that ω ij − ω ′ ij = 0 for j ≤ m while ω ij − ω ′ ij = 0 for j > m (as follows fromProposition 9.3). Hence we have( m + n )( m + n − − m ( m −
1) = n ( n + 2 m − OPOLOGY OF PARAMETRISED MOTION PLANNING ALGORITHMS 21 nonzero classes of degree k − ∗ . Note that the maximal degree ofthe class (9.7) is (2 n + m − k −
1) and thus at best we may hope to have 2 n + m − T ⊂ { , , . . . , m } be a subset and let p > m . Consider the following cohomology classes in H ∗ ( E × B E ):Ω T = Y i ∈ T ω ip and Ω ′ T = Y i ∈ T ω ′ ip . The classes Ω T and Ω ′ T are not among the generators of Proposition 9.3 but theycan be explicitly expressed in terms of the generators, see below.From now on we shall assume that the dimension k ≥ ω ij and ω ′ ij is then even and they commute. Besides, weshall formally introduce classes ω ij with i > j based on the convention ω ij = − ω ji . Lemma 9.5.
One has the identities Ω T = ( − | T |− · X i ∈ T Y j ∈ T −{ i } ω ij · ω ip (9.9) and Ω ′ T = ( − | T |− · X i ∈ T Y j ∈ T −{ i } ω ij · ω ′ ip . (9.10) Proof.
The formula is obviously true for | T | = 1 and we shall assume by inductionthat it is true for all subsets of cardinality smaller than | T | . Let r be the maximalelement of T and let T ′ = T − { r } . Applying our induction hypothesis and the three term relation we getΩ T = Ω T ′ · ω rp = ( − | T | X i ∈ T ′ Y j ∈ T ′ −{ i } ω ij · ω ip · ω rp = ( − | T | X i ∈ T ′ Y j ∈ T ′ −{ i } ω ij · ω ir ( ω rp − ω ip )= ( − | T |− X i ∈ T ′ Y j ∈ T −{ i } ω ij · ω ip + ( − | T | X i ∈ T ′ Y j ∈ T −{ i } ω ij · ω rp . Applying the induction hypothesis again we get Y s ∈ T ′ ω sr = ( − | T | X i ∈ T ′ Y j ∈ T ′ −{ i } ω ij · ω ir = ( − | T | X i ∈ T ′ Y j ∈ T −{ i } ω ij . Hence the sum in the square brackets equals Y s ∈ T ′ ω sr = ( − | T |− Y j ∈ T −{ r } ω rj . This completes the proof. (cid:3)
Consider the product x = m Y i =2 ( ω i ( m +1) − ω ′ i ( m +1) ) · m + n Y j = m +1 ( ω j − ω ′ j ) . (9.11)It is a product of 2 n + m − x = 0 will allow us to useProposition 7.3 to obtain the lower bound TC [ F ( R k , n + m ) → F ( R k , m )] ≥ n + m − OPOLOGY OF PARAMETRISED MOTION PLANNING ALGORITHMS 23
Since we assume that k ≥ ω ij − ω ′ ij ) = − ω ij ω ′ ij . Hence x = 0 is equivalent to y = 0 where y = m Y i =2 ( ω i ( m +1) − ω ′ i ( m +1) ) · m + n Y j = m +1 ( ω j ω ′ j )= m Y i =2 ( ω i ( m +1) − ω ′ i ( m +1) ) · ( ω m +1) ω ′ m +1) ) · m + n Y j = m +2 ( ω j ω ′ j )= m Y i =2 ( ω i ( m +1) − ω ′ i ( m +1) ) · ( ω m +1) ω ′ m +1) ) · ω IJ ω ′ IJ . Here I = { , , . . . , } , and J = { m + 2 , m + 3 , . . . , m + n } ; note that I is a sequenceof length n − p = m + 1) we may write y = X T,S ( − | S | Ω T · Ω ′ S · ω IJ ω ′ IJ , (9.12)where T, S ⊂ { , , . . . , m } run over all subset satisfying T ∩ S = { } and T ∪ S = { , , . . . , m } . Using the result of Lemma 9.5 we obtain the following expressionfor the class ( − m +1 · y : X T,S ( − | S | X i ∈ T Y j ∈ T − i ω ij · ω i ( m +1) X α ∈ S Y β ∈ S − α ω αβ · ω ′ α ( m +1) ω IJ ω ′ IJ . Expanding the brackets gives a sum of many monomials, and each of these monomi-als is one of the generators described in Proposition 9.3. However some monomialsappear several times and may potentially cancel each other.Consider the monomial − ω ω ω . . . ω m ω m +1) ω ′ m +1) ω IJ ω ′ IJ (9.13)which arises by taking T = { } , S = { , , . . . , m } , i = 1 and α = 2. It is easyto see that this is the only choice producing this monomial. Indeed, if the set T contained an element i > ω i would be present which is not thecase. If the set S would miss certain element j > ω j wouldnot be present, however all such factors are present. Thus, we see that the class y ∈ H ∗ ( E × B E ) is nonzero since it contain as a summand one of the free generatorswhich cannot be cancelled by any other term.This completes the proof of Theorem 9.1. (cid:3) Acknowledgments.
Portions of this work were undertaken when the first and secondauthors visited the University of Florida Department of Mathematics in November,2019. We thank the department for its hospitality and for providing a productivemathematical environment.
References
1. K. Borsuk,
Theory of retracts , Monografie Matematyczne, Tom 44, Pa´nstwowe WydawnictwoNaukowe, Warsaw, 1967; MR0216473.2. A. Dranishnikov, and R. Sadykov,
The topological complexity of the free product.
Math. Z.293 (2019), 407416. MR40022823. R. Engelking,
General Topology , translated from the Polish by the author, second edition,Sigma Ser. Pure Math., 6, Heldermann, Berlin, 1989; MR1039321.4. E. Fadell and L. Neuwirth,
Configuration spaces . Math. Scand. (1962), 111-118.MR01411265. E. Fadell and S. Husseini, Geometry and Topology of Configuration Spaces , Springer Mono-graphs in Mathematics, Springer-Verlag, Berlin, 2001; MR1802664.6. M. Farber,
Topological complexity of motion planning , Discrete Comput. Geom. 29 (2003),no. 2, 211221. MR19572287. M. Farber,
Topology of robot motion planning , in:
Morse Theoretic Methods in Non-linearAnalysis and in Symplectic Topology , pp. 185–230, NATO Science Series II: Mathematics,Physics and Chemistry, vol. 217, Springer, 2006; MR2276952.8. M. Farber, M. Grant,
Topological complexity of configuration spaces , Proc. Amer. Math. Soc. (2009), 1841–1847; MR2470845.9. M. Farber, M. Grant, S. Yuzvinsky,
Topological complexity of collision free motion planningalgorithms in the presence of multiple moving obstacles , in:
Topology and Robotics , pp. 75–83,Contemp. Math., vol. 438, Amer. Math. Soc., Providence, RI, 2007; MR2359030.10. M. Farber, S. Yuzvinsky,
Topological robotics: Subspace arrangements and collision free mo-tion planning , in:
Geometry, topology, and mathematical physics , pp. 145–156, Amer. Math.Soc. Transl. Ser. 2, vol. 212, Amer. Math. Soc., Providence, RI, 2004; MR2070052.11. J.M. Garc´ıa-Calcines,
A note on covers defining relative and sectional categories , TopologyAppl. (2019); MR398263612. M. Grant, G. Lupton, L. Vandembrouq, eds.,
Topological complexity and related topics , Con-temp. Math., vol. 702. Amer. Math. Soc., Providence, RI, 2018; MR3762828.13. J. Gonzlez, B. Gutirrez,
Topological complexity of collision-free multi-tasking motion planningon orientable surfaces . Topological complexity and related topics, 151163, Contemp. Math.,702, Amer. Math. Soc., Providence, RI, 2018. MR376283814. A. Hatcher,
Algebraic topology , Cambridge University Press, Cambridge, 2002; MR186735415. J.-C. Latombe,
Robot Motion Planning , Springer Science+Business Media, New York, 2012.16. S. M. LaValle,
Planning Algorithms , Cambridge University Press, Cambridge, 2006;MR2424564.17. A.S. Schwarz,
The genus of a fiber space , Amer. Math. Sci. Transl., (1966), 49140;MR0154284.18. E. Spanier, Algebraic Topology , corrected reprint, Springer-Verlag, New York-Berlin, 1981;MR0666554.
OPOLOGY OF PARAMETRISED MOTION PLANNING ALGORITHMS 25
19. T. Srinivasan,
The Lusternik-Schnirelmann category of metric spaces , Topology Appl. (2014), 87–95; MR3193429.
Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana70803 USA
E-mail address : [email protected] URL : School of Mathematical Sciences, Queen Mary University of London, E1 4NSLondon
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Department of Mathematics, The University of Chicago, 5734 S University Ave,Chicago, IL 60637
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