Effectual Topological Complexity
Natalia Cadavid-Aguilar, Jesús González, Bárbara Gutiérrez, Cesar A. Ipanaque-Zapata
aa r X i v : . [ m a t h . A T ] F e b Effectual Topological Complexity
Natalia Cadavid-Aguilar ∗ , Jes´us Gonz´alez, B´arbara Guti´errez † and Cesar A. Ipanaque-Zapata ‡ Abstract
We introduce the effectual topological complexity (ETC) of a G -space X .This is a G -equivariant homotopy invariant sitting in between the effective topo-logical complexity of the pair ( X, G ) and the (regular) topological complexityof the orbit space
X/G . We study ETC for spheres and surfaces with antipodalinvolution, obtaining a full computation in the case of the torus. This allowsus to prove the vanishing of twice the non-trivial obstruction responsible forthe fact that the topological complexity of the Klein bottle is 4. In addition,this gives a counterexample to the possibility —suggested in Paveˇsi´c’s work onthe topological complexity of a map— that ETC of (
X, G ) would agree withFarber’s TC ( X ) whenever the projection map X → X/G is finitely sheeted. Weconjecture that ETC of spheres with antipodal action recasts the Hopf invariantone problem, and describe (conjecturally optimal) effectual motion planners. : Primary 55M30. Secondary 57S25, 68T40,93C85.
Keywords and phrases:
Sectional category, motion planning, G -space. For a group G and a G -space X , we define the effectual topological complexity (ETC) TC G effl ( X ) as the sectional category of the fibration ǫ : P X → X × ( X/G ) givenby ǫ ( γ ) = ( γ (0) , [ γ (1)]), where square brackets stand for G -orbits. This definitionis motivated by and closely related to (though different from) B laszczyk-Kaluba’s ∗ The first author is grateful for support from Fundaci´on Sof´ıa Koval´evskaia and from FORDE-CYT grant 265667 “Programa para un avance global e integral de la matem´atica mexicana”. † Supported by the project 20201646 of the Secretar´ıa de Investigaci´on y Posgrado at the IPN. ‡ The author would like to thank grant Cadavid-Aguilar, Gonz´alez, Guti´errez and Ipanaque-Zapata effective topological complexity (etc) TC G effv ( X ), revisited in Section 2. In addition,for nicely behaved G -spaces, ETC is a special case of Paveˇsi´c’s TC ( π ), the topologicalcomplexity of the projection map π : X → X/G . But more importantly, ETC servesas a connecting link between TC G effv ( X ) and Farber’s topological complexity of theorbit space, TC ( X/G ): Theorem 1.1. If X is Hausdorff and G is a discrete group acting properly discon-tinuously on X , then (1) TC G effv ( X ) ≤ TC G effl ( X ) ≤ TC ( X/G ) . Both etc and ETC are motivated by the idea of taking advantage of symmetries inthe motion planning of an autonomous system (see Section 3). Yet the two conceptsare essentially different from each other, as indicated next.The case of an n -dimensional sphere ( n ≥
1) with antipodal involution ( S n , Z )and quotient space P n = S n / Z , the n -dimensional real projective space, is particu-larly interesting. It is easy to see that TC Z effv ( S n ) = 1 (Corollary 2.10). More difficultto prove is TC Z effl ( S n ) ∈ { n, n + 1 } and TC ( S n / Z ) = Imm ( P n ) − , n = 1 , , , otherwise.Here Imm ( P n ) stands for the smallest Euclidean dimension where P n admits a smoothimmersion. The values above for TC Z effv ( S n ) and TC ( S n / Z ) come from [1] and [7],respectively, while the estimate for TC Z effl ( S n ) comes from [9, Proposition 4.7]. Thelatter estimate should be compared to a folk empirical belief that the (currentlyunknown) value of Imm ( P n ) would have the form 2 n − α ( n ) + o ( α ( n )), where α ( n )denotes the number of ones in the dyadic expansion of n .Our current knowledge of Imm ( P n ) (summarized in [5]) gives that, for antipodallyacted spheres, both inequalities in (1) are strict as long as n ≥
8. In contrast, inthe parallelizable sphere case ( n = 1 , , TC Z effl ( S n ) = TC ( S n / Z ) = n , for in fact Imm ( P n ) = n + 1. We believe that these three special values of n are the onlyones having TC Z effl ( S n ) = n , which would yield a new form of the Hopf invariantone problem. Such a possibility should be compared to the (homotopically moreaccessible) fact in [8] that TC ( S n ) is minimal possible precisely when the classicalHopf invariant of the Whitehead square of the identity on S n vanishes (i.e., when n is odd).As with spheres, the behavior of the inequalities in (1) is very subtle in the caseof an orientable closed surface with antipodal involution. Explicitly, think of the ori-entable surface Σ g of genus g embedded in R as shown in Figure 1, so that reflectionsin the xy -, yz -, and xz -planes yield symmetries of Σ g . Let σ stand for the “antipo- · · · · · · xy Figure 1: Embedding Σ g ⊂ R dal” (orientation-reversing) involution on Σ g given by σ ( x, y, z ) = ( − x, − y, − z ). Thismakes Σ g into a Z -space with quotient N g +1 , the nonorientable surface of genus g +1.The case g = 1 is similar to the situation noted above for S n with n ≥
8, in that:
Theorem 1.2.
Both inequalities in (1) are strict for ( X, G ) = (T , Z ) , the 2-toruswith antipodal involution. In fact (2) TC Z effv (T) = 2 , TC Z effl (T) = 3 and TC (T / Z ) = 4 . The values for TC Z effv (T) and TC (T / Z ) come from [2, paragraph following Theo-rem 1.1] and [3], respectively. Unlike the first equality in (2), the proof argument (inSection 4) for the second equality in (2) is far from being elementary; yet it is notas technically involved as Cohen-Vandembroucq’s proof of the third equality in (2).On the other hand, Theorem 1.2 is somehow singular among surfaces for, just as forparallelizable spheres, at least one of the inequalities in (1) is an equality in the caseof a larger genus surface. Indeed, Theorem 1.1, [3] and [2, Theorem 1.1] yield(3) 3 ≤ TC Z effv (Σ g ) ≤ TC Z effl (Σ g ) ≤ TC (Σ g / Z ) = 4 , for g ≥ ,neither in the effective nor in the effectual realms, and it might of course be plausiblethat all the numbers in (3) equal 4. Such a possibility could be addressed via ob-struction theory, though the explicit details would seem to be even more challengingthan those carried out for g = 1 in [3] (note that Σ / Z is the Klein bottle). Let
P X stand for the free-path space of a topological space X . Recall that Farber’stopological complexity TC ( X ) is the sectional category of the end-points evaluationmap e , : P X → X × X . (We use sectional category of a map f : E → B in thereduced sense, i.e., one less than the minimal number of open sets covering B andon each of which f admits a homotopy local section.) Let G be a topological groupacting on the right on X . For an integer k ≥
2, let P Gk ( X ) be the subspace of Cadavid-Aguilar, Gonz´alez, Guti´errez and Ipanaque-Zapata ( P X × G ) k − × P X consisting of the tuples ( α , g , . . . , α k − , g k − , α k ) such that α i (1) · g i = α i +1 (0). Note that P Gk ( X ) sits inside P Gk +1 ( X ) as a subspace retract, withinclusion ι : P Gk ( X ) → P Gk +1 ( X ) and retraction r : P Gk +1 ( X ) → P Gk ( X ) given by ι ( α , g , . . . , g k − , α k ) = ( α , g , . . . , g k − , α k , e, α k (1)) ,r ( α , g , . . . , g k , α k +1 ) = ( α , g , . . . , g k − , α k ) , where α k (1) stands for the path with constant value α k (1). α α α α (1) · g α (1) · g Figure 2: An element of P G ( X ) Lemma 2.1.
The G -twisted evaluation map ε : P X × G → X × X given by ε ( α, g ) =( α (0) , α (1) · g ) is a fibration.Proof. Note that ε is the standard fibrational substitute of the “ G -saturated diagonal”∆ G : X × G → X × X given by ∆ G ( x, g ) = ( x, xg ). Proposition 2.2.
For k ≥ , the evaluation map e k : P Gk ( X ) → X × X given by e k ( α , g , . . . , g k − , α k ) = ( α (0) , α k (1)) is a fibration.Proof. This follows from the commutative diagram with pullback square P Gk ( X ) (cid:31) (cid:127) / / (cid:15) (cid:15) e k ( ( ( P X × G ) k − × P X ε k − × e (cid:15) (cid:15) X × X k − × X (cid:31) (cid:127) × ∆ k − × / / π ,k +1 (cid:15) (cid:15) ( X × X ) k X × X where π ,k +1 projects onto the first and last coordinates. Definition 2.3.
The k -effective topological complexity ( k -etc) of the G -space X ,denoted by TC G,k effv ( X ), is the sectional category of e k .B laszczyk-Kaluba’s k -th effective topological complexity TC G,k ( X ) is defined in [1]as the sectional category of the fibration ε k : P G,k ( X ) → X × X . Here P G,k ( X ) = { ( α , . . . , α k ) ∈ ( P X ) k : α i (1) · G = α i +1 (0) · G for 1 ≤ i < k } and ε k ( α , . . . , α k ) = ( α (0) , α k (1)). As detailed in the next paragraphs, k -etc has aslightly better behavior than B laszczyk-Kaluba’s. For starters, TC G,k effv ( X ) keeps thebasic properties of TC G,k ( X ) (Propositions 2.7 and 2.8 below). In fact, the equality TC G,k effv ( X ) = TC G,k ( X ) holds for reasonably nice G -spaces (Remark 2.4 below). Butmore importantly, k -etc has better conceptual properties than B laszczyk-Kaluba’s(see Propositions 2.5, 2.6 and 3.3 below). Remark 2.4.
An element in P Gk ( X ) is designed to encode precise “leaping” in-formation that assembles a broken path in P G,k ( X ). In particular, the projection P : P Gk ( X ) → P G,k ( X ) that forgets all the “group coordinates” satisfies e k = ε k ◦ P ,so that(4) TC G,k ( X ) ≤ TC G,k effv ( X ) . This inequality is in fact an equality if the G -action on X is principal, for in sucha case P : P Gk ( X ) → P G,k ( X ) is a homeomorphism. (Recall that a free action isprincipal if the map τ : Im(∆ G ) → G satisfying x · τ ( x, y ) = y is continuous, where∆ G is the G -saturated diagonal in the proof of Lemma 2.1.)The restriction of e k +1 to P Gk ( X ) is e k , which readily gives the monotonic behavior(5) TC G,k +1 effv ( X ) ≤ TC G,k effv ( X ) , an analogue of TC G,k +1 ( X ) ≤ TC G,k ( X ) ([1, Lemma 3.2(2)]). While B laszczyk andKaluba’s show the latter inequality to be an equality for principal actions, (5) is sharpunder no special restrictions. Proposition 2.5.
For k ≥ , TC G,k effv ( X ) = TC G,k +1 effv ( X ) . Proof.
In view of (5), it suffices to observe that the map f k : P Gk +1 ( X ) → P Gk ( X ), f k ( α , g , . . . , α k − , g k − , α k , g k , α k +1 ) = ( α , g , . . . , α k − , g k − g k , ( α k · g k ) ⋆α k +1 ), yieldsa commutative diagram P Gk +1 ( X ) f k / / e k +1 (cid:15) (cid:15) P Gk ( X ) e k y y rrrrrrrrrr X × X. Here ⋆ stands for concatenation of paths. Cadavid-Aguilar, Gonz´alez, Guti´errez and Ipanaque-Zapata
It therefore suffices to restrict attention to TC G, effv ( X ) which, from now on, willsimply be denoted by TC G effv ( X ). Proposition 2.6.
Let ε and ∆ G be the G -twisted evaluation map and the G -saturateddiagonal, respectively (see Lemma 2.1 and its proof). Then TC G effv ( X ) = secat ( ε ) = secat (∆ G ) . Proof.
The map φ : P G ( X ) → P X × G given by φ ( α , g, α ) = ( α ⋆ ( α · g − ) , g ) is ahomeomorphism with inverse given by φ − ( α, g ) = ( α ′ , g, α ′′ · g ), where α ′ , α ′′ ∈ P X are given by α ′ ( t ) = α ( t/
2) and α ′′ ( t ) = α ( t ) (see Figure 2). The first asserted α (0) α (1 / α (1) α ′ α ′′ Figure 3: α = α ′ ⋆ α ′′ equality then follows from the commutative triangle P G ( X ) φ / / e (cid:15) (cid:15) P X × G ε x x qqqqqqqqqqq X × X. The second asserted equality comes from the commutative diagram X × G c × / / ∆ G (cid:15) (cid:15) P X × G ε x x qqqqqqqqqqq X × X, where c : X ≃ −→ P X stands for the homotopy equivalence sending x ∈ X into thestationary path x .Proposition 2.7 below is verified along the lines of [6, Theorem 3] (cf. [1, Theo-rem 3.3]), while Proposition 2.8 below is a mild generalization of [1, Lemma 3.2(1)].Both properties are direct consequences of Definition 2.3, so we leave the easy detailsas an exercise for the reader. Proposition 2.7.
Let f : X → Y be a G -map with a (not necessarily equivariant)right homotopy inverse g : Y → X , i.e., f ◦ g ≃ Y . Then TC G effv ( Y ) ≤ TC G effv ( X ) . Inparticular TC G effv is a G -homotopy invariant. Proposition 2.8.
For a G -trivial space X , TC G effv ( X ) agrees with TC ( X ) , Farber’stopological complexity of the underlying space X . Further, for a group morphism κ : G → G and a G -space X , we have TC G effv ( X ) ≤ TC G effv ( X ) , where X is seen asa G -space via κ . In particular, TC G effv ( X ) ≤ TC ( X ) . Proposition 2.8 and (4) yield the inequalities TC G,k ( X ) ≤ TC G effv ( X ) ≤ TC ( X ) forany k ≥ G -spaces. In particular, [1, Proposition 5.3] yields Corollary 2.9below, while [1, Proposition 5.9] and the paragraph following (4) yield Corollary 2.10below. Other calculations in [1, Subsection 5.2] also give full information (gatheredin Corollary 2.11 below) in the TC G effv realm. Corollary 2.9.
Let p be a prime integer. Assume that Z p acts cellularly on a pos-itive dimensional sphere S n . If p = 2 , assume in addition that the action preservesorientation. Then TC Z p effv ( S n ) = , if n is odd ;2 , if n is even . Corollary 2.10.
Let Z be a cellular free action on a positive dimensional sphere S n .Then TC Z effv ( S n ) = 1 . Corollary 2.11.
Let Z act cellularly on S n . Assume the action is orientation-reversing and that it has an r -dimensional fixed point set with ≤ r ≤ n − . Then TC Z effv ( S n ) = 1 provided either n is odd, or n is even and the action is linear.Proof. Proposition 2.8 and [1, Lemma 5.5] give 1 ≤ TC Z , ( S n ) ≤ TC Z effv ( S n ). Theequality 1 = TC Z , ( S n ) is obtained in [1, Proposition 5.6] by constructing an explicit2-ruled motion planner which is effective in their sense. Examination of that con-struction reveals that the motion planner is actually effective in our sense. The resultfollows. As explained in [1], the goal in the effective- TC viewpoint is to motion-plan an au-tonomous system by taking advantage of potential symmetries. Explicitly, we aim atinstructing the system to move from any given initial state to any desired final state,contenting ourselves to arrive at a state that is only G -symmetric to the intendeddestination state. The purpose of this section is to show that the latter task has a Cadavid-Aguilar, Gonz´alez, Guti´errez and Ipanaque-Zapata completely different nature than the task of motion planning from any given initialstate to a desired final G -orbit . The resulting concept, which we call the effectual topological complexity of the system (with respect to the given symmetries), turnsout to be a connecting link in the relationship between the effective topological com-plexity of a G -space X and the usual topological complexity of the orbit space X/G .As we will see, such a relationship is rather subtle even for an orientable surfaceendowed with its standard antipodal involution.
Definition 3.1.
The effectual topological complexity (ETC) of a G -space X , denotedby TC G effl ( X ), is the sectional category of the map ǫ : P X → X × ( X/G ) given by thecomposite
P X e −→ X × X × π −−→ X × ( X/G ) , where π stands for the canonical projection, and e is the end-points evaluation map. Remark 3.2.
As shown in [9, Lemma 4.1 and Corollary 4.2], ǫ is a fibration if andonly if π is so, in which case TC G effl ( X ) is nothing but Paveˇsi´c’s topological complexityof π . It will also be convenient to record that, when π is a fibration and both X and X/G are compact metric ANR’s, the openess requirement for coverings of X × ( X/G )can be waved from the definition of TC G effl ( X ) (cf. [9, Theorem 4.6]). In such a case,an effectual motion planner for X is a partition of X × ( X/G ) by subsets D i (calledthe effectual domains ) together with a family of continuous sections s i : D i → P X (called the effectual instructions ) for the restricted fibrations ǫ i : ǫ − ( D i ) → D i . Theeffectual motion planner is said to be optimal if it has TC G effl ( X ) + 1 effectual domains.In this paper we are mainly interested in free G -spaces X for which π : X → X/G is a covering projection. So, throughout the rest of the paper we assume that X isHausdorff and that the action of G on X is properly discontinuous (and thus principal,see [11, Lemma 14.1.1]). In particular, G is assumed to be discrete. In such cases, [9]gives a thorough study of the basic homotopy properties of ETC. We thus focus onits connections to other TC-invariants: Proposition 3.3.
Both squares in the commutative diagram (6) P G ( X ) q / / e (cid:15) (cid:15) P X
P π / / ǫ (cid:15) (cid:15) P ( X/G ) e (cid:15) (cid:15) X × X × π / / X × ( X/G ) π × / / ( X/G ) × ( X/G ) are strict pullbacks. Here q ( α, g, β ) = α ⋆ ( βg − ) . In particular (1) holds.Proof. Consider the commutative diagram
P X × G P G X P XX × X X × ( X/G ) φ − ε qe ǫ × π where φ is the homeomorphism in the proof of Proposition 2.6, so that the top hor-izontal composite is projection onto the the first coordinate. Let P be the pullbackof 1 × π and ǫ . Note that the canonical map ϕ : P X × G → P is surjective: given( x, y, γ ) ∈ P , so that x = γ (0) and [ y ] = [ γ (1)], say y = γ (1) · g with g ∈ G , wehave ϕ ( γ, g ) = ( x, y, γ ). Since X is G -free, ϕ is injective. Thus the assertion for theleft hand-side square in (6) will follow once we show that ϕ : P X × G → P is atopological embedding.Since G is discrete, P X × G = F g ∈ G P X × { g } has the disjoint-union topology.On the other hand, for each g ∈ G , the commutative diagram P X × { g } P X × G P X × X × P XP X, ϕ π where π is the projection onto the third coordinate, shows that the restriction of ϕ to P X × { g } is a topological embedding. Consequently, it is enough to checkthat P has the disjoint-union topology F g ∈ G I g , where I g is the image of P X × { g } under ϕ . We argue in fact that, in X × X × P X , I g ∩ I g = ∅ for g = g : Assumethere is an element ( γ (0) , γ (1) · g , γ ) ∈ I g ∩ I g , and take a neighborhood W of γ (1) such that W · g ∩ W = ∅ whenever g = e . Since X × ( W · g ) × [ { } , W ]is a neighborhood of ( γ (0) , γ (1) · g , γ ) in X × X × P X , there exists and element( α (0) , α (1) · g , α ) ∈ ( X × ( W · g ) × [ { } , W ]) ∩ I g . Therefore α (1) · g ∈ W · g and α (1) ∈ W , which yields α (1) · g ∈ W · g ∩ W · g , and so g = g .We now deal with the right hand-side square in (6), namely, the exterior commu-tative square in the diagram P X B P ( X/G ) X × ( X/G ) (
X/G ) × ( X/G ) , ψ P πǫ e π × Cadavid-Aguilar, Gonz´alez, Guti´errez and Ipanaque-Zapata where B is the pullback of π × e . The canonical map ψ : P X → B is one-to-one because π is a covering map, so we only need to check continuity of ψ − . Considerthe commutative diagram with pullback square P X Q P ( X/G ) X X/G
P πe ℓ e π where the lifting ℓ for the canonical map P X → Q is continuous in view of [10,Theorem II.7.8] (recall π has been assumed to be a covering projection). In theseconditions, ψ − is continuous as it factors as the composite of the two bottom hori-zontal maps in the commutative diagram X × X/G × P ( X/G ) X × P ( X/G ) B Q P X, π ℓ where π projects onto the the first and third coordinates. Remark 3.4.
Paveˇsi´c shows in [9] that the right hand-side square in (6) is a homo-topic pullback whenever π : X → X/G is a fibration.
We now deal with the middle inequality in (2), i.e.:
Theorem 4.1. TC Z effl (T) = 3 . Before proving this fact, we discuss a couple of important consequences. Firstof all, at the end of [9], Paveˇsi´c suggests the posibility that TC G effl ( X ) = TC ( X ) forevery finitely sheeted covering projection π : X → X/G . Theorem 4.1 gives anexplicit counterexample to such a situation. Secondly, recall that the calculationof the topological complexity of the Klein bottle K in [3] amounts to showing thenon-vanishing of the mod-2 reduction of ν TC K , the primary (and unique) obstructionresponsible for the equality TC (K) = 4. In view of the functoriality of primaryobstructions ([12, Theorem VI.6.3]), Theorem 4.1 and the pullback diagram1 P T P KT × K K × K P πǫ e π × show that ν TC K maps trivially under the map π × × K −→ K × K. These ob-servations lead to the following retrospective explanation of Cohen-Vandembroucq’ssuccessful mod-2 calculations.
Corollary 4.2.
Cohen-Vandembroucq’s non-trivial obstruction ν TC K is 2-torsion.Proof. We recall a few preliminary facts from [4]. Let I stand for the augmentationideal of the fundamental group π (K), i.e., I is the kernel of the augmentation map Z [ π (K)] → Z . The fundamental group π (K × K) = π (K) × π (K) acts on Z [ π (K)]and, by restriction, on I via the formula( a, b ) · Σ n i c i = X n i (cid:16) ac i b − (cid:17) . Lastly, the obstruction ν TC K lies in the twisted cohomology group H (K × K; I ⊗ ).Cohen and Vandembrouq assess ν TC K through its Poincar´e-dual image H (K × K; I ⊗ ) −→ H (cid:16) K × K; I ⊗ ⊗ e Z (cid:17) ν TC K ν TC K ∩ [K × K] , where e Z stands for the orientation module of K × K, and [K × K] is the correspondingtwisted fundamental class—a generator of H (K × K; e Z ) ∼ = Z . A similar obstruction-theory setting holds when the fibration e : P K → K × K is replaced by its pullbackfibration ǫ : P T −→ T × K. In particular, we highlight that the orientation module Z of T × K is the pull back of e Z under π × × K → K × K.As a last preliminary ingredient in the proof, consider the group presentations π (T) = h a, b : ab = ba i and π (K) = h x, y : yxy = x i , with generators chosen so that the covering π : T → K has π ∗ ( a ) = x and π ∗ ( b ) = y (see Figure 4). It is standard that the corresponding elements x , x ∈ π (K × K)act on e Z by interchanging sign, while y , y ∈ π (K × K) act trivially. Subindicesare used to indicate coordinate source. Likewise, a , b , y ∈ π (T × K) act triviallyon Z , while x ∈ π (T × K) acts by sign interchange. With this preparation, adirect group-cohomology calculation (left as an exercise for the reader) shows thatthe induced map( π × ∗ : H (cid:16) T × K; Z (cid:17) ∼ = Z −→ Z ∼ = H (K × K; e Z )2 Cadavid-Aguilar, Gonz´alez, Guti´errez and Ipanaque-Zapata a abbx xy y TK π Figure 4: The double cover of the Klein bottle by the torusis multiplication by −
2, i.e., ( π × ∗ takes the twisted fundamental class [T × K] onto − × K]. Thus (cid:16) ν TC K (cid:17) ∩ [K × K] = ν TC K ∩ (2[K × K])= ν TC K ∩ (cid:16) − ( π × ∗ [T × K] (cid:17) = − ( π × ∗ (cid:16) ( π × ∗ (cid:16) ν TC K (cid:17) ∩ [T × K] (cid:17) , where the latter expression vanishes, since ( π × ∗ (cid:16) ν TC K (cid:17) = 0, as observed at theend of the paragraph following the statement of Theorem 4.1. Poincar´e duality thenyields 2 ν TC K = 0.The rest of the section is devoted to the proof of Theorem 4.1.The following cohomology facts are standard and easy to prove (all cohomologygroups below are taken with mod-2 coefficients). H ∗ (T) is generated by elements α, β ∈ H (T) = Z ⊕ Z subject to the relations α = β = 0 and αβ = γ ,where γ stands for the generator of H (T) = Z . Likewise, H ∗ (K) is generatedby elements κ, λ ∈ H (K) = Z ⊕ Z , subject to the relation κλ = 0 and κ = λ = µ, where µ stands for the generator of H (K) = Z . Furthermore, the map π ∗ : H ∗ (K) → H ∗ (T) is determined by π ∗ ( κ ) = π ∗ ( λ ) = α + β . With this informa-tion, the inequality TC Z effl (T) ≥ p : E → B is bounded from below by the cuplength of elements in the kernel of p ∗ : H ∗ ( B ) → H ∗ ( E ). Explicitly, in our situ-ation, the fibration under consideration is ǫ : P T → T × K which, in terms of thestandard homotopy equivalence T ≃ P T, takes the form (1 , π ) : T → T × K. Then(1 , π ) ∗ ( α ⊗ β ⊗ ⊗ λ ) = α + β + ( α + β ) = 0 , whereas a direct calculationyields ( α ⊗ β ⊗ ⊗ λ ) = ( α + β ) ⊗ µ = 0.More difficult is establish TC Z effl (T) ≤
3. Rather than using obstruction theory,we actually describe an (optimal) effectual motion planner with 4 domains, i.e., a3partition of T × K into 4 subsets (not necessarily open, in view of Remark 3.2), eachadmitting a section for the corresponding restriction of ε : P T → T × K.We start by fixing some notation. Think of T as T = S × S , where the first(second) S -coordinate will be depicted horizontally (vertically). In these terms, theantipodal involution on T becomes σ ( x , x ) = ( − x , x ), where z stands for thecomplex conjugate of z ∈ S . For a point x = ( x , x ) ∈ T, set V x := { x } × S ( H x := S × { x } ), the “vertical” (“horizontal”) circle passing through x . V x H x x xy z Set in addition:• M x := n e iθ x : − π ≤ θ ≤ π o × S , the half handle determined by x .• C Ix := n e − iπ x o × S , the “left” boundary component of M x .• C Dx := n e iπ x o × S , the “right” boundary component of M x .• C x := ( S × {− x } ) ∩ M x .• A x := C Ix ∪ C x ∪ C Dx .• x ′ := ( x , − x ) , the “antipodal to x in V x ”.• a x := (cid:16) e iπ x , − x (cid:17) , b x := (cid:16) e − iπ x , − x (cid:17) , so C Ix ∩ C x = { b x } and σ ( a x ) = b x .4 Cadavid-Aguilar, Gonz´alez, Guti´errez and Ipanaque-Zapata C Dx C Ix C x x x ′ M xπ − π xy z Finally set a := H (1 , − = S × {− } ( b := H (1 , = S × { } ), the “inner” (“outer”)horizontal circle. Note that both a and b are closed under the involution.We are ready to define the domains D i ( i = 1 , , ,
4) and corresponding sectionsthat complete the proof of Theorem 4.1. The first domain is D := { ( x, z ) ∈ T × K : there exists y ∈ M x \ A x with z = π ( y ) } . Note that σ ( y ) / ∈ M x if y ∈ M x \ A x . Therefore the π -preimage y of z in the definitionof D is unique. Furthermore, such an element y ∈ π − ( z ) ∩ ( M x \ A x ) clearly dependscontinuously on ( x, z ) ∈ D . Thus, a section s : D → P T as the one we need sendsa pair ( x, z ) ∈ D into the path in T from x to y depicted by the thick arrows inFigure 5, i.e., we first adjust the first S -coordinate, and then adjust the second S -coordinate. The continuity on x and y of these adjustments comes from the factsthat y ∈ M x (for the first adjustment) and y / ∈ A x (for the second adjustment).Note that T × K \ D = { ( x, z ) : π − ( z ) ∩ A x = ∅ } . The second domain is D := { ( x, z ) : x / ∈ a ∪ b and there exists y ∈ A x \ (cid:16) C Ix ∪ { a x } (cid:17) with π ( y ) = z } . As in the case of D , the π -preimage y of z in the definition of D is unique anddepends continuously on ( x, z ) ∈ D . Since C Dx is closed, the continuity assertionis not completely obvious when y ∈ C Dx \ { a x } and, for such a case, we offer thefollowing argument. Let V be a small neighborhood of y , say small enough so that V and σ ( V ) are contained in different sides of the hyperplane P x in R spannedby V x (see Figure 6). Since σ ( y ) = b x (for y = a x ), we can shrink V if neededso to assume that σ ( V ) does not intersect some tubular neighborhood τ of C x (seeFigure 6). Since x ′ , C x and C Dx depend continuously on x , there is a neighborhood5 xy y yyyy Figure 5: Motion planning in D C Ix C Dx b x τσ ( V ) yVa x x x ′ P x Figure 6: Neighborhoods V , σ ( V ) and τU of x such that, for any x ∈ U , C x ⊆ τ and C Dx lies on the same side of P x as C Dx does. Under such conditions, for an element ( x , z ) ∈ ( U × π ( V )) ∩ D , the element y ∈ A x \ (cid:16) C Ix ∪ { a x } (cid:17) satisfying π ( y ) = z will also satisfy y ∈ V ∪ σ ( V ) and,for the required continuity, we need to make sure that in fact y ∈ V . Assume, fora contradiction, that y ∈ σ ( V ). Given the choosing of V , it follows that y ∈ C x .So y ∈ C x ⊆ τ , which contradicts τ ∩ σ ( V ) = ∅ . Having established the fact that y ∈ (cid:16) A x \ ( C Ix ∪ a x ) (cid:17) ∩ π − ( z ) depends continuously on ( x, z ) ∈ D , the rest is easy:A section s : D → P T as the one we need sends a pair ( x, z ) ∈ D into the pathin T from x to y depicted by the thick arrows in Figure 7. Note that the condition x / ∈ a ∪ b implies that a x is not the intersection of the half horizontal arc C x with the6 Cadavid-Aguilar, Gonz´alez, Guti´errez and Ipanaque-Zapata [ a x ] yyyy yy x Figure 7: Motion planning in D b x x Figure 8: Motion planning in D vertical circle C Dx , so that the motion planning is precisely as indicated in Figure 7.Note that T × K \ ( D ∪ D ) consists of the pairs ( x, z ) satisfying one of the fol-lowing two conditions:(i) x / ∈ a ∪ b and z = [ a x ].(ii) x ∈ a ∪ b and π − ( z ) ∩ A x = ∅ .The third domain combines the pairs satisfying (i) with some of the pairs satisfy-ing (ii). Let D consist of the pairs ( x, z ) ∈ T × K satisfying (i) above. Since[ a x ] = [ b x ], a section s : D → P T as the one we need sends ( x, [ a x ]) ∈ D intothe path in T from x to b x depicted by the thick arrows in Figure 8. On the other hand,let D consist of the pairs ( x, z ) in (ii) for which there exists y ∈ A x \ (cid:16) C Ix ∪ { a x } (cid:17) with π ( y ) = z . An argument identical to the one given in the case of D showsthat the element y on the definition of D is unique and depends continuously on( x, z ) ∈ D . Thus, a section s : D → P T as the one we need sends ( x, z ) ∈ D x yy yy Figure 9: Motion planning in D into the path in T from x to y depicted by the thick arrows in Figure 9. Note that thecondition x ∈ a ∪ b implies that a x is the intersection point (removed from Figure 9)of C x and C Dx , which yields continuity on ( x, z ) of s . Since an element ( x, z ) ∈ D must have z = [ a x ] , while an element ( x, z ) ∈ D must have x ∈ a ∪ b , it follows that D ∩ D = ∅ = D ∩ D . Therefore s and s yield a section s on D := D ∪ D as the one we need.The last domain is D := T × K − ( D ∪ D ∪ D ) , i.e., D consist of the pairs ( x, [ a x ]) ∈ ( a ∪ b ) × K. As in the case of D , D admits asection s : D → P T as the one we need. The proof of Theorem 4.1 is now complete.
We close the paper by describing an effectual motion planner on S n ⊂ R n +1 with n +2domains in the general case, and n + 1 domains provided n ∈ { , , } . We conjecturethat these are optimal planners, i.e., that TC Z effl ( S n ) = n + δ n , where δ n = 1 exceptfor δ = δ = δ = 0.Set k = n + δ n and choose a continuous map v = ( v , v , . . . , v k ) : S n → ( S n ) k +1 with v ( p ) = p and so that(7) v ( p ) , v ( p ) , . . . , v k ( p ) generate R n +1 for each p ∈ S n .For instance, in the non-parallelizable case, v i ( p ) can be taken to be the i -th canonicalbasis element (for all p ). For p ∈ S n , let H i ( p ) = { q ∈ R n +1 : h q, v i ( p ) i = 0 } and H + i ( p ) = { q ∈ R n +1 : h q, v i ( p ) i > } , where h− , −i denotes the standard inner productin R n +1 . Recall that π : S n → P n stands for the projection. The sets D i = ( p, ℓ ) ∈ S n × P n : ℓ ∈ π \ ≤ j
Publicacions Matematiques , 62:55–74, 2018.[2] Natalia Cadavid-Aguilar and Jes´us Gonz´alez. Effective topological complexityof orientable-surface groups, arXiv:1907.10212v2, 2020. To appear in
Topologyand its applications. [3] Daniel C. Cohen and Lucile Vandembroucq. Topological complexity of the kleinbottle.
Journal of Applied and Computational Topology , 1:199–213, 2017.[4] Armindo Costa and Michael Farber. Motion planning in spaces with small funda-mental groups.
Communications in Contemporary Mathematics ∼ dmd1/imms.html.[6] Michael Farber. Topological complexity of motion planning. Discrete Comput.Geom. , 29(2):211–221, 2003.[7] Michael Farber, Serge Tabachnikov, and Sergey Yuzvinsky. Topological robotics:Motion planning in projective spaces.
International Mathematics Research No-tices , 2003(34):1853–1870, 2003.[8] Jes´us Gonz´alez, Mark Grant, and Lucile Vandembroucq. Hopf invariants, topo-logical complexity, and LS-category of the cofiber of the diagonal map for two-cellcomplexes. In
Topological complexity and related topics , volume 702 of
Contemp.Math. , pages 133–150. Amer. Math. Soc., Providence, RI, 2018.[9] Petar Paveˇsi´c. Topological complexity of a map.
Homology, Homotopy andApplications , 21(2):107–130, 2019.[10] Edwin H. Spanier.
Algebraic topology . McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966.9[11] Tammo tom Dieck.
Algebraic topology . EMS Textbooks in Mathematics. Euro-pean Mathematical Society (EMS), Z¨urich, 2008.[12] George W. Whitehead.
Elements of homotopy theory , volume 61 of
GraduateTexts in Mathematics . Springer-Verlag, New York-Berlin, 1978.
Departamento de Matem´aticasCentro de Investigaci´on y de Estudios Avanzados del I.P.N.Av. Instituto Polit´ecnico Nacional no. 2508, San Pedro ZacatencoM´exico City 07000, M´exico [email protected]@math.cinvestav.mx
Departamento de Formaci´on B´asica DisciplinariaUnidad Profesional Interdisciplinaria de Ingenier´ıa Campus HidalgoCarretera Pachuca-Actopan Km. 1+500Ciudad del Conocimiento y la Cultura 42162, Hidalgo [email protected]