An upper bound for topological complexity
aa r X i v : . [ m a t h . A T ] J u l AN UPPER BOUND FOR TOPOLOGICAL COMPLEXITY
MICHAEL FARBER, MARK GRANT, GREGORY LUPTON, AND JOHN OPREA
Abstract.
In [11], a new approximating invariant TC D for topologicalcomplexity was introduced called D -topological complexity. In this pa-per, we explore more fully the properties of TC D and the connectionsbetween TC D and invariants of Lusternik-Schnirelmann type. We alsointroduce a new TC -type invariant f TC that can be used to give an upperbound for TC , TC ( X ) ≤ TC D ( X ) + (cid:24) X − kk + 1 (cid:25) , where X is a finite dimensional simplicial complex with k -connecteduniversal cover ˜ X . The above inequality is a refinement of an estimategiven by Dranishnikov [5]. Introduction
Topological complexity TC ( X ) is a numerical homotopy invariant intro-duced by Farber [8]. As well as being of intrinsic interest to homotopy the-orists, its study is motivated by topological aspects of the motion planningproblem in robotics. The number TC ( X ) gives a quantitative measure ofthe ‘navigational complexity’ of X , when viewed as the configuration spaceof a mechanical system. Topological complexity is a close relative of theLusternik–Schnirelmann category cat ( X ) (see [1]), although the two notionsare independent.Recall that cat ( X ) is the smallest n such that X admits an open covering { U , . . . , U n } by ( n +1) sets, each of which is contractible in X . The sectionalcategory of a fibration p : E → B , denoted by secat ( p ), is the smallest number n for which there is an open covering { U , . . . , U n } of B by ( n + 1) open sets,for each of which there is a continuous local section s i : U i → E of p ; thatis, p ◦ s i = j i : U i → B , where j i denotes the inclusion.Let X I denote the space of (free) paths in a space X . There is a fibration π X : X I → X × X, which evaluates a path at initial and final points: for α ∈ X I , we have π X ( α ) = (cid:0) α (0) , α (1) (cid:1) . We define the topological complexity TC ( X ) of X tobe the sectional category secat (cid:0) π X (cid:1) of this fibration. That is, TC ( X ) is the Mathematics Subject Classification.
Primary 55M30; Secondary 55P99.
Key words and phrases. topological complexity, Lusternik-Schnirelmann category.This work was partially supported by a grant from the Simons Foundation: ( smallest number n for which there is an open cover { U , . . . , U n } of X × X by( n +1) open sets, for each of which there is a continuous section s i : U i → X I of π X , π X ◦ s i = j i : U i → X × X , where j i denotes the inclusion.Just as LS category is very difficult to compute, so also is topologicalcomplexity. Indeed, it is usually the case for both invariants that lower andupper bounds are derived. The fundamental such bounds are the following. Theorem 1.
The following bounds hold [8] : cat ( X ) ≤ TC ( X ) ≤ cat ( X × X ) . When X = K ( π,
1) is aspherical the topological complexity TC ( X ) de-pends only on π and we may write TC ( X ) = TC ( π ). It is easy to see (usingthe Eilenberg - Ganea theorem [7]) that TC ( π ) is finite if and only if thereexists a finite dimensional K ( π, Theorem 2.
Let X be a finite CW-complex with fundamental group π .Then TC ( X ) ≤ TC ( π ) + dim( X ) . (1)Of course, this estimate is only meaningful when TC ( π ) is finite. This rulesout, for instance, any group π having torsion. The inequality (1) is effectivewhen the group π has “small” cohomological dimension, say, is trivial, afree group or a surface group etc.In this paper, we will refine the estimate of Theorem 2 by using an in-variant TC D ( X ) defined in [11] as well as a new invariant f TC ( X ) defined inthis article (see Example 3.6). Theorem 3.
Let X be a CW-complex with fundamental group π . Then TC ( X ) ≤ TC D ( X ) + f TC ( X ) . (2) Moreover, if the universal cover ˜ X is k -connected, then TC ( X ) ≤ TC D ( X ) + (cid:24) X − kk + 1 (cid:25) . (3)The definitions of the invariants TC D ( X ) and f TC ( X ) are given in § §
3. Note, however, that in contrast to TC ( π ), TC D ( X ) is finite whenever TC ( X ) is. Furthermore, it equals TC ( π ) for X = K ( π, TC D ( X ) ≤ TC ( π ) , where π = π ( X ) . (4)For the invariant f TC ( X ), we will see that it is a special case of a new typeof sectional category invariant ] secat ( E p → X q → X ) associated to a coveringmap q : X → X and a fibration p : E → X . The invariant ] secat ( E p → X q → X )and its properties will be explored in §
3. As a special case of Theorem 4.1,we will also see that f TC ( X ) ≤ dim( X ) . (5) N UPPER BOUND FOR TOPOLOGICAL COMPLEXITY 3
Hence both terms on the right-hand side of inequality (2) are dominatedby the corresponding terms of the right-hand side of (1). We give specificexamples when (2) is sharper than (1). We note, however, that Dranish-nikov actually proves a stronger inequality than (1) using his notion of strongly equivariant topological complexity TC ∗ π and uses the more “prac-tical” inequality (1) because TC ∗ π is very difficult to compute. We show inProposition 3.8 that our invariant f TC ( X ) is equal to TC ∗ π ( ˜ X ) so that in thegeneral case (2) is a refinement because of the first term TC D ( X ) alone.Nevertheless, using f TC provides not only a much simpler proof of the gen-eral upper bound in Theorem 3 but also allows the generalization to theimproved connectivity-dimension upper bound in Theorem 3 as well.2. The D -Topological Complexity Let us recall from [11] the following definition.
Definition 2.1.
Let X be a path-connected topological space with fundamen-tal group π = π ( X ; x ) . The D -topological complexity, TC D ( X ) , is definedas the minimal number k such that X × X can be covered by k + 1 opensubsets X × X = U ∪ U ∪ · · · ∪ U k with the property that for any i = 0 , , , . . . , k and for every choice of thebasepoint u i ∈ U i , the homomorphism π ( U i ; u i ) → π ( X × X ; u i ) inducedby the inclusion U i → X × X takes values in a subgroup conjugate to thediagonal ∆ ⊂ π × π . Note that the letter D in the notation TC D ( X ) stands for the “diagonal”.Here we mention that for each point u i ∈ X × X , there is an isomorphism π ( X × X ; u i ) → π ( X × X ; ( x ; x )) = π × π determined uniquely up toconjugation, and the diagonal inclusion X → X × X induces the inclusion π → π × π onto the diagonal ∆.Recall that a topological space X admits a universal cover if it is con-nected, locally path connected and semi-locally simply connected. Sincethese conditions are preserved under products, it then follows that X × X admits a universal cover. In particular, X × X admits a universal coverwhenever X is a locally finite cell complex. Proposition 2.2.
Let X be a connected, locally path connected and semi-locally simply connected topological space with fundamental group π = π ( X ; x ) .Let q : \ X × X → X × X be the connected covering space corresponding to thediagonal subgroup ∆ ⊂ π × π = π ( X × X ; ( x ; x )) . Then the D -topologicalcomplexity satisfies TC D ( X ) = secat ( q ); that is, TC D ( X ) equals the sectional category of the covering q .Proof. If U ⊂ X × X is an open subset, then a partial section U → \ X × X of q gives a factorisation of the homomorphism of fundamental groups induced MICHAEL FARBER, MARK GRANT, GREGORY LUPTON, AND JOHN OPREA by the inclusion U → X × X through the diagonal. Now, since q is acovering, for an open subset U ⊂ X × X , the condition that the inducedmap π ( U ; u ) → π ( X × X ; u ) takes values in a subgroup conjugate to thediagonal ∆ implies that q admits a continuous section over U . Using thisremark the result follows by comparing the definitions of TC D ( X ) and ofsectional category. (cid:3) Example . For a path-connected space X one has TC D ( X ) = 0 if andonly if X is simply connected; this follows directly from the definition. Inparticular we have TC D ( S n ) = 0 for all n >
1. Also, we have that TC D ( S ) =1 as follows from TC D ( S ) > TC D ( S ) ≤ TC ( S ) = 1 (see Proposition 2.4 below).Next we compare TC D ( X ) with TC ( X ). Proposition 2.4.
For a connected, locally path connected and semi-locallysimply connected topological space X one has TC D ( X ) ≤ TC ( X ) . Proof.
The following argument appears in the proof of Theorem 4.1 of [12].Let ˜ X → X be the universal cover of X . Consider the projection Q :˜ X × π ˜ X → X × X where π = π ( X ) denotes the fundamental group of X and ˜ X × π ˜ X stands for the quotient of ˜ X × ˜ X with respect to the diagonalaction of π . Clearly Q is a covering map with the property that the image ofthe induced homomorphism Q ∗ : π ( ˜ X × π ˜ X ) → π ( X × X ) is the diagonal.Hence by Proposition 2.2 one has TC D ( X ) = secat ( Q ).Define p : X I → ˜ X × π ˜ X by p ( γ ) = [˜ γ (0) , ˜ γ (1)], where ˜ γ : I → e X is anylift of the path γ : I → X and the brackets [ x, y ] denote the orbit of thepair ( x, y ) ∈ ˜ X × ˜ X with respect to the diagonal action of π . The map p is well-defined although of course the lift ˜ γ is not unique. We obtain thefollowing commutative diagram. X I p / / π X ●●●●●●●●● ˜ X × π ˜ X Q y y ssssssssss X × X Clearly, a partial section s : U → X I gives a partial section ˜ s = p s : U → ˜ X × π ˜ X , so we have TC ( X ) = secat ( π X ) ≥ secat ( Q ) = TC D ( X ) . (cid:3) Proposition 2.5.
Let X be an aspherical locally finite cell complex. Then TC D ( X ) = TC ( X ) . N UPPER BOUND FOR TOPOLOGICAL COMPLEXITY 5
Proof.
Recall the known fact that an open subset of a locally finite cellcomplex is homotopy equivalent to a countable cell complex. Indeed, byTheorem 1 of [21], a space is homotopy equivalent to a countable cell complexif and only if it is homotopy equivalent to an absolute neighbourhood retract(ANR). Any locally finite cell complex is an ANR and an open subset of anANR is an ANR [16]. Thus, an open subset of a locally finite cell complexis an ANR and hence has the homotopy type of a countable cell complex.In view of Proposition 2.4 we only need to establish the inequality TC ( X ) ≤ TC D ( X ). Consider an open subset U ⊂ X × X such that the map inducedby the inclusion U → X × X on fundamental groups takes values in a sub-group conjugate to the diagonal. Since X × X is aspherical and U has thehomotopy type of a cell complex, we see that the inclusion U → X × X is homotopic to a map with values in the diagonal ∆ X ⊂ X × X . Nowwe can use Lemma 4 from [10] to conclude that a section of the path fi-bration π X : X I → X × X over U exists. The statement follows from thedefinitions. (cid:3) Proposition 2.6.
Let f : X → Y be a continuous map between path-connected topological spaces such that the induced map f ∗ : π ( X ) → π ( Y ) is an isomorphism. Then TC D ( X ) ≤ TC D ( Y ) . Proof.
Let U ⊂ Y × Y be an open subset such that the induced homomor-phism π ( U, u i ) → π ( Y × Y, u i ) takes values in a subgroup conjugate to thediagonal ∆ Y . Consider the preimage V = ( f × f ) − ⊂ X × X . The homo-morphism π ( V, v i ) → π ( X × X, v i ) induced by the inclusion V → X × X takes values in a subgroup conjugate to the diagonal ∆ X . Hence any opencover of Y × Y as in Definition 2.1 defines a similar covering on X × X withthe same number of sets. (cid:3) Corollary 2.7. TC D ( X ) is a homotopy invariant of X . We may therefore write TC D ( π ) = TC D ( K ( π, TC D ( π ) = TC ( π ). Corollary 2.8.
Let X be a path-connected cell complex with fundamentalgroup π = π ( X ) . Then TC D ( X ) ≤ TC D ( π ) . (6) Moreover, if π has cohomological dimension ≤ , TC D ( X ) = TC D ( π ) . (7) Proof.
First note that we may construct the Eilenberg–Mac Lane complex K = K ( π,
1) starting from X and attaching cells of dimension ≥
3. We mayapply Proposition 2.6 to the inclusion X ⊂ K which obviously induces anisomorphism of fundamental groups; this gives inequality (6).To prove (7) we first convert the inclusion X ֒ → K into a fibration withfibre F satisfying π i ( F ) = π i +1 ( K, X ). Note that, since K is aspherical and MICHAEL FARBER, MARK GRANT, GREGORY LUPTON, AND JOHN OPREA π ( X ) ∼ = π ( K ), we have π i ( F ) = π i +1 ( K, X ) = 0 for i = 0 ,
1. Now, theobstructions to finding a section of the fibration X → K lie in the groups(with local coefficients) H i +1 ( K ; π i ( F )) = H i +1 ( π ; π i ( F )). But by what wehave said above, these are trivial for i = 0 ,
1. Furthermore, the hypothesisthat cd( π ) ≤ H i +1 ( π ; π i ( F )) = 0 for i >
1. Hence, allobstructions vanish and there is a section K → X . In particular, the sectionis an isomorphism on π since X → K is, so we simply apply Proposition 2.6to get TC D ( π ) = TC D ( K ) ≤ TC D ( X ) . Combining this with the first part then gives equality. (cid:3)
As a generalisation of the previous Corollary we state:
Lemma 2.9.
Let X be a path-connected cell complex such that for someinteger k ≥ the homotopy groups π j ( X ) are zero for all j satisfying 1) can be ob-tained from X by attaching cells of dimension k + 1 , k + 2 , . . . . We have X ⊂ K with π i +1 ( K, X ) = 0 for i = 0 , , . . . , k − X → K with fibre F satisfying π i ( F ) = 0for i = 0 , , . . . , k − 1. The obstructions to finding a section of X → K liein the groups H i +1 ( π, π i ( F )) = H i +1 ( K, π i ( F )) and all these groups van-ish because of our computation with π i ( F ) and our assumption cd( π ) ≤ k .Finally we apply Proposition 2.6 to achieve equality. (cid:3) Example . Let X be a finite cell complex with fundamental group π .Suppose that cd( π ) > X . Then we have TC D ( X ) ≤ TC ( X ) ≤ X < cd( π ) ≤ TC D ( π ) . Thus one may construct many examples with TC D ( X ) < TC D ( π ). For in-stance, we may take a 2-dimensional finite cell complex X with fundamentalgroup Z (i.e. the 2-skeleton of T ). Furthermore, since every finitely pre-sented group π appears as the fundamental group of a closed 4-manifold X ,the gap between TC D ( X ) and TC D ( π ) = TC ( π ) can be as large as desired.Let X be a connected, locally path connected and semi-locally simply con-nected space with universal cover P : e X → X . The Lusternik-Schnirelmann one-category , cat ( X ), is defined as the sectional category secat ( P ) of P .This interpretation of one-category goes back to Schwarz ([26]) who alsoshowed that cat ( X ) = cat ( f ), where f : X → K ( π ( X ) , 1) classifies the uni-versal cover and cat ( f ) is the category of the map f (also see [23]). Thislatter description easily implies, for instance, that cat ( T n × Y ) = n when Y is simply connected. We describe now a relation between cat ( X ) and TC D ( X ) that is akin to that between cat ( X ) and TC ( X ). N UPPER BOUND FOR TOPOLOGICAL COMPLEXITY 7 Proposition 2.11. If X is a connected, locally path connected and semi-locally simply connected topological space then cat ( X ) ≤ TC D ( X ) ≤ cat ( X × X ) . (8) Proof. Consider first the following general situation. Let p : ˜ Z → Z be acovering map with ˜ Z connected and p ∗ ( π ( ˜ Z, ˜ z )) = H ⊂ π ( Z, z ). Let f : A → Z be an inclusion of a connected subspace. We obtain a pull-backdiagram ˜ Z A / / p A (cid:15) (cid:15) ˜ Z p (cid:15) (cid:15) A f ⊂ / / Z in which ˜ Z A can be identified with the preimage of A under p and p A isthe restriction of p . Clearly the map p A is a covering map. The set ˜ Z A isconnected if and only if f ∗ ( π ( A )) and p ∗ ( π ( ˜ Z )) span π ( Z ). In that case p A is a connected covering corresponding to the subgroup f − ∗ ( H ) ⊂ π ( A ).Now consider the following diagram. X / / p ′ (cid:15) (cid:15) \ X × X q (cid:15) (cid:15) ˜ X × ˜ X o o P × P (cid:15) (cid:15) X f ⊂ / / X × X X × X = o o (9)Here P : ˜ X → X is the universal cover of X and q : \ X × X → X × X isthe cover corresponding the diagonal subgroup ∆ ⊂ π × π . The map f is aninclusion f ( x ) = ( x, x ), where x ∈ X and x ∈ X is a base-point and X isthe preimage q − ( f ( X )). To apply the remark of the preceding paragraph,note that f ∗ ( π ( X )) and q ∗ ( π ( \ X × X )) span π ( X × X ). Hence it followsthat p ′ : X → X is the universal cover of X .Given an open subset U ⊂ X × X with a section s : U → \ X × X wemay restrict it to f − ( U ) ⊂ X getting a section s ′ : f − ( U ) → X . Thisshows that cat ( X ) = secat ( p ′ ) ≤ secat ( q ) = TC D ( X ), thus proving the leftinequality (8).Next we consider the right square of the diagram (9). The map P × P isthe universal covering and hence secat ( P × P ) = cat ( X × X ) ≥ secat ( q ) = TC D ( X ). (cid:3) Next we examine the case of real projective spaces. Proposition 2.12. For any n ≥ one has TC D ( RP n ) = TC ( RP n ) . MICHAEL FARBER, MARK GRANT, GREGORY LUPTON, AND JOHN OPREA Hence, TC D ( RP n ) = n for n = 1 , , and for any n = 1 , , the number TC D ( RP n ) equals the smallest integer k = k ( n ) such that the projective space RP n admits an immersion into R k .Proof. Assume first that n = 1 , , 7. In this case we may combine Theorem4.1, Corollary 4.4, Propositions 6.2 and 6.3 from [12] which imply our state-ment for n = 1 , , 7. In the remaining case, i.e. when n = 1 , , 7, we knowby [12] that TC ( RP n ) = n = cat ( RP n ) = cat ( RP n ) ≤ TC D ( RP n ) . (Note that, in [12], the non-normalised convention TC ( ∗ ) = 1 was usedrather than the normalized convention TC ( ∗ ) = 0 of this paper.) The state-ment now follows from Proposition 2.4. (cid:3) Remark . In [12, Theorem 4.5] it is shown that for n = 2 k , TC ( RP n ) =2 k +1 − 1. When k = 3, for instance, we see that the lowest immersiondimension for RP is 15, so that TC D ( RP ) = 15, a result that would bedifficult to obtain directly from the definition.The next two results are analogues of results that equate TC to Lusternik-Schnirelman category for certain types of spaces. Although the first resultfollows from the more general second, the relative simplicity of the proof inthe presence of greater structure recommends its inclusion. Proposition 2.14. For any connected topological group G one has TC D ( G ) = cat ( G ) . Proof. Let F : G × G → G be the map given by the formula F ( a, b ) = ab − .Denote π = π ( G, e ) and consider the induced map on fundamental groups φ = F ∗ : π × π = π ( G × G, e × e ) → π. We claim that the kernel of φ coincides with the diagonal subgroup ∆ ⊂ π × π . Obviously the kernel of φ contains ∆. On the other hand, standardEckmann-Hilton arguments used in the proof that π ( G ) is abelian showthat F ∗ ( a, b ) = a − b , so that K er ( φ ) = ∆.This gives a pullback diagram of covering maps \ G × G q (cid:15) (cid:15) ˜ F / / ˜ G P (cid:15) (cid:15) G × G F / / G where P is the universal covering and q is the covering corresponding to thediagonal subgroup. From this diagram we obtain TC D ( G ) = secat ( q ) ≤ secat ( P ) = cat ( G ) . This complements the left inequality of Proposition 2.11. (cid:3) Next we give a generalisation of Proposition 2.14. N UPPER BOUND FOR TOPOLOGICAL COMPLEXITY 9 Theorem 2.15. Let X be a connected CW H -space. Then TC D ( X ) = cat ( X ) . Proof. The proof given below is based on arguments used in [20] to showthat TC ( X ) = cat ( X ) when X is an H -space.Let m : X × X → X denote the multiplication, which may be assumedto have a strict unit given by the base point x ∈ X . If A is a based CWcomplex and f, g : A → X are based maps, their pointwise product f · g : A → X is defined by ( f · g )( a ) = m ( f ( a ) , g ( a )) for all a ∈ A . By a theorem ofJames [19], the pointwise product endows the set of based homotopy classes[ A, X ] with the structure of an algebraic loop . In particular, equations ofthe form x · a = b, a · y = b, a, b ∈ [ A, X ]admit unique solutions x, y ∈ [ A, X ].Let p , p : X × X → X denote the coordinate projections. The loopstructure on [ X × X, X ] guarantees the existence of a difference map D : X × X → X with the property that p · D ≃ p : X × X → X .We claim that the induced homomorphism D ∗ : π ( X × X ) = π ( X ) × π ( X ) → π ( X ) on fundamental groups is given by D ∗ ( a, b ) = b − a forall a, b ∈ π ( X ). To see this, recall that the standard proof that π ( X ) isabelian when X is an H -space proceeds by showing that the two binaryoperations + : π ( X ) × π ( X ) → π ( X ) and · : π ( X ) × π ( X ) → π ( X ),given respectively by concatenation and pointwise product of loops, share atwo-sided identity and are mutually distributive. Therefore they agree. Itfollows that a + D ∗ ( a, b ) = ( p ) ∗ ( a, b ) + D ∗ ( a, b )= ( p ) ∗ ( a, b ) · D ∗ ( a, b )= ( p · D ) ∗ ( a, b )= ( p ) ∗ ( a, b )= b, which proves the claim.Now form the pullback of the universal cover p : e X → X along the map D : X × X → X to obtain a covering ρ : P → X × X with secat ( ρ ) ≤ secat ( p ) = cat ( X ). The image ρ ∗ π ( P ) in π ( X × X ) is contained in thekernel of D ∗ , which as we have just seen equals the diagonal subgroup. Hencethere is a lift P → \ X × X of ρ through the covering q : \ X × X → X × X ,which gives TC D ( X ) = secat ( q ) ≤ secat ( ρ ).Combining the two inequalities above, we have that TC D ( X ) ≤ cat ( X )when X is an H -space. The opposite inequality is given by Proposition 2.11. (cid:3) In the somewhat overlooked paper [14], D. Handel shows that the free pathfibration X I → X × X is a pullback of the based path fibration P X → X over a certain map h : X × X → X (which we describe below) if and onlyif X is a CW H-space. This in itself implies TC ( X ) = cat ( X ) by standardinequalities, but it also can be used to give another proof of Theorem 2.15.Both proofs use special properties of CW H-spaces, so it is worthwhile seeingeach of them. Alternative Proof of Theorem 2.15. We use the notation of the proof of The-orem 2.15. The loop structure of [ X, X ] shows that there is a right inversemap η : X → X with the property that X ∆ → X × X X × η −−−→ X × X m → X is nullhomotopic. Then we define h = m (1 X × η ). On homology, this mapinduces h ∗ ( a, b ) = a − b and since, for an H-space, π ( X ) = H ( X ), we seethat h ( a, b ) = a − b as well. Then, since the diagonal ∆( π ) ⊂ π × π is thekernel of this homomorphism, we can lift h to ˜ h : \ X × X → ˜ X as in \ X × X ˜ h / / q (cid:15) (cid:15) ˜ X P (cid:15) (cid:15) X × X h / / X. But the homomorphism h induces an isomorphism ( π × π ) / ∆( π ) ∼ = π andthis in turn shows that ˜ h restricted to each fibre ( π × π ) / ∆( π ) → π is abijection. This then implies that the diagram above is a pullback and theusual sectional category inequality gives TC D ( X ) = secat ( q ) ≤ secat ( P ) = cat ( X ) . (cid:3) Fibration over a Covering and the Invariant f TC ( X )Consider the situation E p → X q → X where p is a fibration with fibre F and q is a covering map where the space X is connected. The composition q ◦ p : E → X is a fibration with fibre F ′ which is homeomorphic to the product F × F where F is the fibre of q , i.e. a discrete set. Definition 3.1. Define the number ] secat( E p → X q → X ) as the minimalinteger k ≥ such that X admits an open cover X = U ∪ · · · ∪ U k withthe property that for each i = 0 , , . . . , k the fibration p admits a continuoussection over q − ( U i ) ⊂ X . N UPPER BOUND FOR TOPOLOGICAL COMPLEXITY 11 We see immediately from the definition that ] secat ( E p → X q → X ) ≥ secat ( p ) , (10)and ] secat ( E p → X q → X ) = 0 if and only if the fibration p : E → X admits acontinuous section, i.e. when secat ( p ) = 0. Presently, we have no exampleswhere the inequality (10) is strict. We then obtain the following estimate. Proposition 3.2. One has secat ( q ◦ p ) ≤ secat ( q : X → X ) + ] secat ( E p → X q → X ) . (11)We postpone the proof to recall some known results about open covers.These results are described and proved in [3, 4]; the other relevant referencesare [25, 15, 2, 22] as well as [1, Exercise 1.12].An open cover W = { W , . . . , W m + k } of a space X is an ( m + 1) -cover if every subcollection { W j , W j , . . . , W j m } of m + 1 sets from U also covers X . The following simple observation (see [25] for instance) is often givenwithout proof, but it is the basis for many arguments in this approach. Lemma 3.3. A cover W = { W , W , . . . , W k + m } is an ( m + 1) -cover of X if and only if each x ∈ X is contained in at least k + 1 sets of W .Proof. If W is an ( m + 1)-cover and x ∈ X is only in k sets in W , then k + m + 1 − k = m + 1 sets of the cover do not contain x . These m + 1 setsdo not cover X , contradicting the supposition on W .Suppose that each x ∈ X is contained in at least k + 1 sets from W andchoose a subcollection V of m + 1 sets from W . There are only k + m + 1 − ( m + 1) = k sets not in V , so x must belong to at least one set in V . Thus V covers X , and W is an ( m + 1)-cover. (cid:3) An open cover can be lengthened to a ( k +1)-cover, while retaining certainessential properties of the sets in the cover: Theorem 3.4 ([2, 3]) . Let U = { U , . . . , U k } be an open cover of a normalspace X . Then, for any m = k, k + 1 , . . . , ∞ , there is an open ( k + 1) -coverof X , { U , . . . , U m } , extending U such that for n > k , U n is a disjoint unionof open sets that are subsets of the U j , ≤ j ≤ k . In Theorem 3.4, the sets U n possess any properties of the original coverthat are inherited by disjoint unions and open subsets. In particular, ifthe cover U is categorical, then the extended cover is also categorical. Thefollowing was proved in [22]. We recall the proof for the convenience of thereader. Lemma 3.5. Let X be a normal space with two open covers U = { U , U , . . . , U k } and V = { V , V , . . . , V m } such that each set of U satisfies Property (A), and each set of V satisfiesProperty (B). Assume that Properties (A) and (B) are inherited by open subsets and disjoint unions. Then X has an open cover W = { W , W , . . . , W k + m } by open sets satisfying both Property (A) and Property (B).Proof. Using Theorem 3.4, extend U to a ( k + 1)-cover e U = { U , . . . , U k + m } and extend V to an ( m + 1)-cover e V = { V , . . . , V k + m } . Since each set in e U is a disjoint union of open subsets of sets in U , the cover e U consists of setssatisfying Property (A); likewise, each set in e V satisfies Property (B). SinceProperties (A) and (B) are inherited by open subsets and disjoint unions,we see that each set U i ∩ V j satisfies both properties.Therefore, the lemma will be proved if we can show that the collection W = { U ∩ V , U ∩ V , . . . , U k + m ∩ V k + m } . is an open cover of X . First, observe that since e V is an ( m + 1)-cover, eachpoint x ∈ X lies in at least k + 1 sets of e V ; we may suppose, without lossof generality, that x ∈ V ∩ · · · ∩ V k . Next, since e U is a ( k + 1)-cover, thesubcollection { U , . . . , U k } covers X , and so x ∈ U i for some 0 ≤ i ≤ k .Thus x ∈ U i ∩ V i for at least one value of i and W covers X . (cid:3) Proof of Proposition 3.2. We say that an open subset U ⊂ X satisfies prop-erty A if q has a section over U . We say that an open subset U ⊂ X satisfiesproperty B if the fibration p has a section over q − ( U ) ⊂ X . Both propertiesA and B are inherited by subsets and disjoint unions. If k = secat ( q ) and l = ] secat ( E p → X q → X ) then there exists an open cover of X by k + 1 opensets satisfying A and there exists an open cover of X by l + 1 open subsetssatisfying B. Hence by Lemma 3.5 there is an open cover of X by k + l + 1open subsets satisfying A and B . If U ⊂ X is an open subset satisfying Aand B then q has a section over U and p has a section over q − ( U ). Then q ◦ p has a section over U . Hence secat ( q ◦ p ) ≤ k + l . (cid:3) Example . Let E = { ( x, y, ω ); x, y ∈ ˜ X, ω ∈ ˜ X I , ω (0) = x, ω (1) = y } /π = X I , where π = π ( X ). Let X = ˜ X × π ˜ X . Let p : E → X be given by p ([ x, y, ω ]) =[ x, y ] and let q : X → X × X be given by q ([ x, y ]) = ( P x, P y ), where P : ˜ X → X is the universal cover. Now we have the situation X I p → ˜ X × π ˜ X q → X × X. (12)Obviously secat ( q ◦ p ) = TC ( X ) and secat ( q ) = TC D ( X ). We shall introducethe shorthand notation f TC ( X ) = ] secat ( X I p → ˜ X × π ˜ X q → X × X ) . Inequality (11) in this particular case becomes TC ( X ) ≤ TC D ( X ) + f TC ( X ) . (13)This then establishes the first part of Theorem 3 of the Introduction. N UPPER BOUND FOR TOPOLOGICAL COMPLEXITY 13 Remark . In [24] the invariant f cat ( X ) (called universal cover category )was defined to be the least k such that there exist open sets U , . . . , U k whoseunion covers X and whose preimages P − ( U j ) under the universal coveringmap P : ˜ X → X are contractible in ˜ X . It was then shown that the estimate cat ( X ) ≤ cat ( X ) + f cat ( X )(14)holds. In light of Definition 3.1, we see that (since the based path space P ˜ X is contractible) f cat ( X ) = ] secat ( P ˜ X p → ˜ X P → X )with cat ( X ) = secat ( P ◦ p ) and cat ( X ) = secat ( P ). Hence, by Proposition 3.2,(14) is a specialization of (11). Proposition 3.8. For any locally finite cell complex X , the number f TC ( X ) = ] secat ( X I p → ˜ X × π ˜ X q → X × X ) coincides with the strongly equivariant topological complexity TC ∗ π ( ˜ X ) intro-duced by A. Dranishnikov [5] .Proof. Recall that TC ∗ π ( ˜ X ) is defined as the minimal number k such that˜ X × ˜ X can be covered by k + 1 open sets ˜ U i such that each ˜ U i is π × π -invariant and admits a π -equivariant continuous section ˜ s i : ˜ U i → ˜ X I . Let P : ˜ X → X denote the universal covering projection. Each π × π -invariantopen set ˜ U i ⊂ ˜ X × ˜ X has the form ( P × P ) − ( U i ) where U i = ( P × P )( ˜ U i )is an open subset of X × X .The definition of f TC ( X ) deals with open subsets U i ⊂ X × X and con-tinuous sections s i : V i = q − ( U i ) → X I . If ˜ U i ⊂ ˜ X × ˜ X denotes ( P × P ) − ( U i ) then V i equals the quotient ˜ U i /π with respect to the diagonal copy of π ⊂ π × π . We have the commutativediagram ˜ X IP (cid:15) (cid:15) ˜ p / / ˜ X × ˜ X Q (cid:15) (cid:15) ˜ U i ⊃ o o Q | Vi (cid:15) (cid:15) X I p / / ˜ X × π ˜ X V i ⊃ o o (15)in which every vertical arrow is a principal π -bundle.It is clear that every π -equivariant section ˜ s : ˜ U i → ˜ X I of the map˜ p : ˜ X I → ˜ X × ˜ X determines (by passing to π -orbits) the map s : V i → X I ,and since ˜ p ◦ ˜ s is the inclusion ˜ U i → ˜ X × ˜ X we obtain that p ◦ s is theinclusion V i → ˜ X × π ˜ X , i.e. s is a section of p . Thus, TC ∗ π ( ˜ X ) ≥ f TC ( X ).For the converse, we want to show that each section s : V i → X I of p determines a π -equivariant section of ˜ p . To do so we need to recall afew basic facts about principal bundles, which can be found for example in[13, 18]. Each principal π -bundle p : E → B over a space with the homotopy typeof a CW complex is classified by a homotopy class ξ ∈ [ B, Bπ ]. Note that E has a free π -action and B = E/π . If p ′ : E ′ → B ′ is another principal π -bundle with class ξ ′ ∈ [ B ′ , Bπ ] then a morphism of π -bundles E ′ p ′ (cid:15) (cid:15) F / / E p (cid:15) (cid:15) B ′ f / / B exists if and only if f ∗ ( ξ ) = ξ ′ . Here the word morphism means that F : E ′ → E is a continuous map commuting with the π -action. Note also that F is uniquely determined by f up to principal bundle equivalence, in thefollowing sense: if F , F : E ′ → E are two π -maps with p ◦ F i = f ◦ p ′ for i = 0 , F = F ◦ u where u : E ′ → E ′ is a principal bundle equivalence of p ′ , that is, a π -homeomorphism which induces the identity on B ′ .Let s : V i → X I be a continuous section of p . Let ξ ∈ [ ˜ X × π ˜ X, Bπ ]denote the class of the bundle Q , see diagram (15). Then ξ | V i ∈ [ V i , Bπ ]is the class of the bundle Q | V i and η = p ∗ ( ξ ) ∈ [ X I , Bπ ] is the class of thebundle P . One has s ∗ ( η ) = s ∗ p ∗ ( ξ ) = ξ | V i and applying the general theory of principal bundles as described above wesee that s extends to a morphism of principal bundles˜ X IP (cid:15) (cid:15) ˜ U i ˜ s o o Q | Vi (cid:15) (cid:15) X I V is o o Note that ˜ s : ˜ U i → ˜ X I is a π -equivariant map, but need not be a section of˜ p . Consider the composition of morphisms of principal bundles˜ X × ˜ X P × P (cid:15) (cid:15) ˜ X IP (cid:15) (cid:15) ˜ p o o ˜ U i ˜ s o o Q | Vi (cid:15) (cid:15) ˜ X × π ˜ X X Ip o o V is o o We know that the lower map p ◦ s : V i → ˜ X × π ˜ X is the inclusion. Let j : ˜ U i → ˜ X × ˜ X denote the inclusion, which is π -equivariant and covers p ◦ s .Using the uniqueness property of principal bundle maps described above,we see that ˜ p ◦ ˜ s ◦ u = j for some principal bundle equivalence u of Q | V i .Thus ˜ s ◦ u : ˜ U i → ˜ X I is a π -equivariant section of ˜ p . (cid:3) N UPPER BOUND FOR TOPOLOGICAL COMPLEXITY 15 Corollary 3.9. Let P : ˜ X → X be the universal covering projection with π = π ( X ) . Then f TC ( X ) ≥ TC ( ˜ X ) . Proof. Since the definition of strongly equivariant topological complexityjust puts extra conditions on the usual TC diagram for ˜ X , we have TC ∗ π ( ˜ X ) ≥ TC ( ˜ X ). The result follows directly then from Proposition 3.8. (cid:3) Lemma 3.10. A CW complex X is aspherical if and only if f TC ( X ) = 0 .Proof. Suppose that f TC ( X ) = 0; that is, the fibration p : X I → ˜ X × π ˜ X has a continuous section. For n ≥ π n ( X ) = π n ( X I ) p ∗ → π n ( ˜ X × π ˜ X ) ≃ → π n ( X × X ) = π n ( X ) ⊕ π n ( X ) . Since p has a section this composition must be surjective. On the otherhand, it is obvious that the image of this composition coincides with thediagonal π n ( X ) ⊂ π n ( X ) ⊕ π n ( X ). This is possible only when π n ( X ) = 0for all n ≥ X is aspherical so that ˜ X is contractible.The fibre of p : X I → ˜ X × π ˜ X is Ω ˜ X , the loop space of the universalcover, so it is also contractible. This implies that p has a section and hence f TC ( X ) = 0. (cid:3) Proposition 3.11. Let Z = X × Y where X = K ( π, is aspherical and Y is simply connected. Then TC D ( Z ) = TC ( X ) and f TC ( Z ) = TC ( Y ) . Proof. The first statement follows by applying Proposition 2.6 to the maps X → X × Y → X (injection and projection).Now let’s consider f TC ( Z ). The tower of fibrations (12) looks in this caseas follows X I × Y I p X × p Y / / ( ˜ X × π ˜ X ) × ( Y × Y ) q X × q Y / / ( X × X ) × ( Y × Y ) . As we mentioned in the proof of Lemma 3.10, since X is aspherical, thereexists a continuous section σ : ˜ X × π ˜ X → X I , i.e. p X ◦ σ = 1. Let U ⊂ Y × Y be an open set admitting a section s : U → Y I of p Y . Then theset ( q X × q Y ) − ( X × X × U ) = ˜ X × π ˜ X × U admits the section σ × s of p X × p Y . This shows that f TC ( Z ) ≤ TC ( Y ) . Conversely, assume that V ⊂ ( X × X ) × ( Y × Y ) is an open subset suchthat there is a continous section σ : ( q X × q Y ) − ( V ) → X I × X I . Fix a point x ∈ X and denote V ′ = V ∩ ( x × x × Y × Y ) ⊂ Y × Y. Thenclearly there exists a continuous section σ ′ : V ′ → Y I . This shows that TC ( Y ) ≤ f TC ( Z ) . (cid:3) Note that the inequality (13) reduces in this special case (i.e. when Z = X × Y with X aspherical and Y simply connected) to the usual productinequality, TC ( Z ) ≤ TC D ( Z ) + f TC ( Z ) = TC ( X ) + TC ( Y ) . Finally, we give an example showing that Theorem 3 can be sharp andcan be a better estimate than that of Theorem 2 even when TC ( π ) is finite. Example . Consider the product Z = T × S . Then TC D ( Z ) = TC ( T ) = 2 and f TC ( Z ) = TC ( S ) = 2. By applying the zero-divisors-cup-length estimate it is easy to see that TC ( Z ) ≥ TC ( Z ) ≤ 4. Thus (13) is, in thisinstance, an equality.4. A Connectivity-Dimensional Upper Bound In this section we establish an upper bound on ] secat ( E p → X q → X ) interms of the dimension of X and the connectivity of the fibre of p . Theorem 4.1. Let X be a finite dimensional simplicial complex. Considertwo maps E p → X q → X, where q : X → X is a covering map (not necessary regular) and p : E → X is a fibration with ( k − -connected fibre for some k ≥ . Then ] secat ( E p → X q → X ) ≤ (cid:24) dim( X ) − kk + 1 (cid:25) . (16) Proof. First we want to rephrase Definition 3.1 as follows. We claim thatthe number ] secat ( E p → X q → X )equals the smallest c such that there exists an increasing sequence of closedsubsets T − = ∅ ⊂ T ⊂ T ⊂ T ⊂ · · · ⊂ T c = X such that for any i = 0 , , . . . , c the fibration p admits a continuous sectionover the set q − ( T i − T i − ). This follows by repeating the arguments of theproof of Proposition 4.12 from [9]. (It can also be shown below for skeletausing the fact that skeletal pairs are NDR pairs.)Consider the sequence of skeleta X ( k ) ⊂ X (2 k +1) ⊂ X (3 k +2) ⊂ · · · ⊂ X (( c +1) k + c ) = X where c is the smallest integer with ( c + 1) k + c ≥ dim( X ), i.e. c = (cid:24) dim( X ) − kk + 1 (cid:25) . Denoting T i = X (( i +1) k + i ) for i = 0 , , . . . , c , we obtain an increasing se-quence of closed subsets T ⊂ T ⊂ · · · ⊂ T c = X . We want to show N UPPER BOUND FOR TOPOLOGICAL COMPLEXITY 17 that the fibration p : E → X admits a continuous section over the set q − ( T i − T i − ) ⊂ X for each i .Next we make the following remark. Let p : E → X be a fibration withfibre F . Suppose that F is ( k − A ⊂ X be a subset whichis homotopy equivalent to a simplicial complex of dimension ≤ k . Then p admits a section over A . This follows directly by applying obstructiontheory.Applying the result of § T i = X (( i +1) k + i ) , thedifference T i − T i − is homotopy equivalent to a simplicial complex of di-mension ≤ k . Then its preimage q − ( T i − T i − ) is also homotopy equivalentto a simplicial complex of dimension ≤ k . Our statement now follows as weassume that the fibre F of p : E → X is ( k − (cid:3) Theorem 4.1 gives that ] secat ( E p → X q → X ) ≤ dim( X )assuming the fibre of p : E → X is not empty; this is the case k = 0. If thefibre F is connected then ] secat ( E p → X q → X ) ≤ (cid:24) dim( X ) − (cid:25) , and so on. Example . We have that f TC ( RP n ) = ] secat (( RP n ) I p → S n × π S n q → RP n × RP n ) . Because RP n × RP n is covered by the ( n − S n × S n ,Theorem 4.1 gives f TC ( RP n ) ≤ (cid:24) · n − ( n − n − 1) + 1 (cid:25) = 2 . If n is even, then by Corollary 3.9, we have f TC ( RP n ) ≥ TC ( S n ) = 2, so f TC ( RP n ) = 2.Finally we combine the inequality (13) with the upper bound of Theorem4.1 in the situation of Example 3.6. We obtain the following result which isthe second part of Theorem 3 of the Introduction. Theorem 4.3. Let X be a finite dimensional simplicial complex. Assumethat the universal cover ˜ X of X is k -connected, i.e. π i ( ˜ X ) = 0 for i ≤ k .Then TC ( X ) ≤ TC D ( X ) + (cid:24) X − kk + 1 (cid:25) . (17)In the special case k = 1 (which is satisfied without extra assumptions ofconnectivity) the inequality (17) gives TC ( X ) ≤ TC D ( X ) + dim X (18) which can be compared with the result of A. Dranishnikov [5]. The inequal-ity (18) is stronger than ([5]) in cases when TC D ( X ) < TC ( π ).As another special case of Theorem 4.3 we see that assuming that ˜ X is2-connected, one has TC ( X ) ≤ TC D ( X ) + (cid:24) 23 (dim X − (cid:25) . Appendix: Complements For convenience of the reader we state here a few known results which areused in the previous sections.For a simplicial complex K we denote by V ( K ) the set of its vertices.The symbol | K | denotes the geometric realisation of K . Lemma 5.1. Let L ⊂ K be a simplicial subcomplex. Suppose that L hasthe following convexity property: every simplex of K with all its vertices in L lies in L . Then the complement | K | − | L | is homotopy equivalent to asimplicial complex X with the vertex set V ( X ) = V ( K ) − V ( L ) ; a set ofvertices in V ( X ) forms a simplex in X if and only if it forms a simplex in K .Proof. For a vertex v ∈ V ( K ) − V ( L ) we denote by S v ⊂ | K | its open star,i.e. the union of all open simplices containing v . The family { S v ; v ∈ V ( K ) − V ( L ) } is an open cover of the complement | K | − | L | (here we use our assumptionconcerning the convexity of L ). The nerve of this open cover is isomorphicto X , as a simplicial complex. We observe that each nonempty intersection S v ∩ S v ∩ · · · ∩ S v k is contractible since it coincides with the open star ofthe simplex ( v , v , . . . , v k ) ∈ K . Our statement now follows from Corollary4G.3 on page 459 of Hatcher [17]. (cid:3) Next we remove the convexity assumption: Lemma 5.2. For any simplicial subcomplex L ⊂ K , the complement | K | −| L | is homotopy equivalent to the simplicial complex Y with the vertex setlabelled by the set of simplices of K which are not in L ; simplices of Y are in1-1 correspondence with increasing chains σ ( σ ( · · · ( σ k of simplicesof K − L .Proof. Consider the barycentric subdivision K ′ of K . Its vertices are labelledby the simplices of K ; the simplices of K ′ are labelled by the increasingchains σ ( σ ( · · · ( σ k of simplices of K . The barycentric subdivision L ′ of L is a simplicial subcomplex of K ′ . The subcomplex L ′ ⊂ K ′ hasthe convexity property: a simplex of K ′ belongs to L ′ if and only if all itsvertices lie in L ′ . Now we simply apply the previous Lemma. (cid:3) N UPPER BOUND FOR TOPOLOGICAL COMPLEXITY 19 Corollary 5.3. For a d -dimensional simplicial complex K and an integer r < d , the complement | K | − | K ( r ) | has the homotopy type of a simplicialcomplex of dimension ≤ d − r − .Proof. Applying the previous Lemma we see that the complement | K | −| K ( r ) | has homotopy type of a simplicial complex with vertex set labelledby the set of simplices of K having dimension > r and with the simplices in1-1 correspondence with the increasing chains of simplices of dimension > r .Since the length of any such chain is at most d − r the result follows. (cid:3) This Corollary appeared in [23] and also in [6].6. 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School of Mathematical Sciences, Queen Mary University of London, Lon-don, E1 4NS, United Kingdom E-mail address : [email protected] Institute of Pure and Applied Mathematics, University of Aberdeen, Ab-erdeen AB24 3UE, United Kingdom E-mail address : [email protected] Department of Mathematics, Cleveland State University, Cleveland OH44115, U.S.A. E-mail address : [email protected] Department of Mathematics, Cleveland State University, Cleveland OH44115, U.S.A. E-mail address ::