NORIO IWASE AND MICHIHIRO SAKAI

Transcription

1 TOPOLOGICAL COMPLEXITY IS A FIREWISE L-S CATEGORY NORIO IWASE AND MICHIHIRO SAKAI Abstract. Topological complexity TC() of a space is introduced by M. Farber to measure how much complex the space is, which is first considered on a configuration space of a motion planning of a robot arm. We also consider a stronger version TC M () of topological complexity with an additional condition: in a robot motion planning, a motion must be stasis if the initial and the terminal states are the same. Our main goal is to show the equalities TC() = cat * (d()) + 1 and TCM () = cat (d()) + 1, where d() = is a fibrewise pointed space over whose projection and section are given by p d() = pr 2 : the canonical projection to the second factor and s d() = : the diagonal. In addition, our method in studying fibrewise L-S category is able to treat a fibrewise space with singular fibres. 1. Introduction We say a pair of spaces (X, A) is an NDR pair or A is an NDR subset of X, if the inclusion map is a (closed) cofibration, in other words, the inclusion map has the (strong) Strøm structure (see page 22 in G. Whitehead [24]). When the set of the base point of a space is an NDR subset, the space is called well-pointed. Let us recall the definition of a sectional category (see James [14]) which is originally defined and called by Schwarz genus. Definition 1.1 (Schwarz [21], James [15]). For a fibration p : E X, the sectional cateory secat(p) (= one less than the Schwarz genus Genus(p)) is the minimal number m 0 such that there exists a cover of X by (m+1) open subsets U i X each of which admits a continuous section s i : U i E. The topological complexity of a robot motion planning is first introduced by M. Farber [2] in 2003 to measure the discontinuity of a robot motion planning algorithm searching also the way to minimise the discontinuity. At a more general view point, Farber defined a numerical invariant TC() of any topological space : let P() be the space of all paths in. Then there is a Serre path fibration π : P() given by π(`) = (`(0),`(1)) for ` P(). Definition 1.2 (Farber). For a space, the topological complexity TC() is the minimal number m 1 such that there exists a cover of by m open subsets U i each of which admits a continuous section s i : U i P() for π : P(). y definition, we can observe that the topological complexity is nothing but the Schwartz genus or the sectional category. Date: October 27, 2008, [First draft] Mathematics Subject Classification. Primary 55M30, Secondary 55Q25. Key words and phrases. Toplogical complexity, Lusternik-Schnirelmann category. 1

2 2 IWASE AND SAKAI Farber has further introduced a new invariant restricting motions by giving two additional conditions on the section s : U P(). (1) s(b, b) =c b the constant path at b for any b, (2) s(b 1,b 2 )=s(b 2,b 1 ) 1 if (b 1,b 2 ) U. It gives a stronger invariant than the topological complexity, and the Z/2-equivariant theory must be applied as in Farber-Grant [4]. This new topological invariant, in turn, suggests us another motion planning under the condition that a motion is stasis if the initial and the terminal states are the same. Let us state more precisely. Definition 1.3. For a space, the monoidal topological complexity TC M () is the minimal number m 1 such that there exists a cover of by m open subsets U i () each of which admits a continuous section s i : U i P() for the Serre path fibration π : P() satisfying s i (b, b) =c b for any b. Remark 1.4. This new topological complexity TC M is not a homotopy invariant, in general. However, it is a homotopy invariant if we restrict our working category to the category of a space such that the pair (, ()) is NDR. On the other hand, a fibrewise pointed L-S category of a fibrewise pointed space is introduced and studied by James-Morris [13]. Let us recall the definition: Definition 1.5 (James-Morris [13]). (1) Let X be a fibrewise pointed space over. The fibrewise pointed L-S category cat (X) is the minimal number m 0 such that there exists a cover of X by (m + 1) open subsets U i s X () each of which is fibrewise null-homotopic in X by a fibrewise pointed homotopy. If there are no such m, we say cat (X) =. (2) Let f : Y X be a fibrewise pointed map over. The fibrewise pointed L-S category cat (f) is the minimal number m 0 such that there exists a cover of Y by (m + 1) open subsets U i s Y (), where the restriction f Ui to each subset is fibrewise compressible into s X () in X by a fibrewise pointed homotopy. If there are no such m, we say cat (f) =. To describe our main result, we further introduce a new unpointed version of fibrewise L-S category: the fibrewise L-S category cat ( ) of an fibrewise unpointed space is also defined by James and Morris [13] as the minimum number (minus one) of open subsets which cover the given space and are fibrewise null-homotopic (see also James [14] and Crabb-James [1]). In this paper, we give a new version of a fibrewise unpointed L-S category of a fibrewise pointed space as follows: Definition 1.6. (1) Let X be a fibrewise pointed space over. The fibrewise unpointed L-S category cat * (X) is the minimal number m 0 such that there exists a cover of X by (m+1) open subsets U i each of which is fibrewise compressible into s X () in X by a fibrewise homotopy. If there are no such m, we say cat * (X) =. (2) Let f : Y X be a fibrewise pointed map over. The fibrewise unpointed L-S category cat * (f) is the minimal number m 0 such that there exists a cover of Y by (m + 1) open subsets U i, where the restriction f Ui to each subset is fibrewise compressible into s X () in X by a fibrewise homotopy. If there are no such m, we say cat * (f) =. For a given space, we define a fibrewise pointed space d() by d() = with p d() = pr 2 : and s d() = : the diagonal. One of our main goals of this paper is to show the following theorem.

3 TOPOLOGICAL COMPLEXITY IS A FIREWISE L-S CATEGORY 3 Theorem 1.7. For a space, we have the following equalities. (1) TC() = cat * (d()) + 1. (2) TC M () = cat (d()) + 1. Farber and Grant has also introduced lower bounds for the topological complexity by using the cup length and category weight (see Rudyak [17] for example) on the ideal of zero-divisors, i.e, the kernel of : H ( ; R) H (; R). Definition 1.8 (Farber [2] and Farber-Grant [4]). For a space and a ring R 3 1, the zero-divisors cup-length Z R () and the TC-weight wgt π (u; R) for u I = ker : H ( ; R) H (; R) is defined as follows. (1) Z R () = Max {m 0 H ( ; R) I m 6=0} (2) wgt π (u; R) = Max {m 0 f : Y (secat(f π) <m), f (u) = 0} In the category T of fibrewise pointed spaces with base space and maps between them, we also have corresponding definitions. Definition 1.9. For a fibrewise pointed space X over and a ring R 3 1 and u I = H (X, ; R) H (X; R), we define (1) cup (X; R) = Max {m 0 {u 1,,u m H (X, ; R)} s.t. u 1 u m 6=0} o (2) wgt nm 0 (u; R) = Max f : Y X T (cat (f) <m), f (u) = 0 This immediately implies the following. Theorem For a space, we have Z R () = cup (d(); R) for a ring R 3 1. Motivating by this equality, we proceed to obtain the following result. Theorem For any space, any element u H (, (); R) and a ring R 3 1, we have wgt π (u; R) = wgt (u; R). Let us consider one technical condition on a fibrewise pointed space: Theorem For any space having the homotopy type of a locally finite simplicial complex, we may assume that d() is fibrewise well-pointed up to homotopy. The following is the main result of our paper. Theorem For any fibrewise well-pointed space X over, we have cat (X) = cat * (X). So, if is a locally finite simplicial complex, we have TC() =TCM () = cat (d()) + 1. In [19], Sakai showed, in his study of the fibrewise pointed L-S category of a fibrewise well-pointed spaces, using Whitehead style definition, that we can utilise A methods used in the study of L-S category (see Iwase [7, 8]). Let us state the Whitehead style definitions of fibrewise L-S categories following [19]. Definition Let X be a fibrewise well-pointed space over. The fibrewise pointed L-S category cat (X) is the minimal number m 0 such that the (m+1)- fold fibrewise diagonal m+1 : X m+1 Π X is compressible into the fibrewise fat wedge m+1 T X in T. If there are no such m, we say cat (X) =.

4 4 IWASE AND SAKAI We remark that this new definition coincides with the ordinary one, if the total space X is a finite simplicial complex. The above Whitehead-style definition allows us to define the module weight, cone length and categorical length, and moreover, to give their relationship as in Section 8. To show that, we need a criterion given by fibrewise A structure on the fibrewise loop space (see Sections 6 7). 2. Proof of Theorem 1.7 First, we show the equality TC M () = cat (d())+1: assume TCM () =m+1, m 0 and that there are an open cover S m i=0 U i = and a series of sections s i : U i P() of π : P() d() satisfying s i (b, b) =c b for b, since we are considering monoidal topological complexity. Then each U i is fibrewise compressible relative to () into () = d() by a homotopy H i : U i [0, 1] given by the following: H i (a, b; t) = (s i (a, b)(t),b), (a, b) U i,t [0, 1], where we can easily check that H i gives a fibrewise compression of U i relative to () into (). Since S i=0 U i = = d(), we obtain cat (d()) m, and hence we have cat (d()) + 1 T CM (). S Conversely assume that cat (d()) = m, m 0 and there is an open cover m i=0 U i = d() of d() = where U i is fibrewise compressible relative to () into () d() = : let us denote the compression homotopy of U i by H i (a, b; t) = (σ i (a, b; t),b) for (a, b) U i and t [0, 1], where σ i (a, b; 0) = a and σ i (a, b; 1) = b. Hence we can define a section s i : U i P() by the formula s i (a, b)(t) =σ i (a, b; t) t [0, 1]. Since S i=0 U i =, we obtain TC M () m+1 and hence we have TC M () cat (d()) + 1. Thus we have TCM () = cat (d()) + 1. Second, we show the equality TC() = cat * (d()) + 1: assume TC() =m+1, m 0 and that there is a open cover S m i=0 U i = and a section s i : U i P() of π : P() d(). Then each U i is fibrewise compressible into () = d() by a homotopy H i : U i [0, 1] which is given by H i (a, b; t) = (s(a, b)(t),b), (a, b) U i,t [0, 1], where we can easily check that H gives a fibrewise compression of U i into () = d(). Since S i=0 U i = = d(), we obtain cat * (d()) m, and hence we have cat * (d()) + 1 T C(). S Conversely assume that cat * (d()) = m, m 0 and there is an open cover m i=0 U i = d() of d() = where U i is fibrewise compressible into () = d(): the compression homotopy is described as H i (a, b; t) = (σ i (a, b; t),b) for (a, b) U i and t [0, 1], such that σ i (a, b; 0) = a and σ i (a, b; 1) = b. Hence we can define a section s i : U i P() by the formula s i (a, b)(t) =σ i (a, b; t) t [0, 1]. Since S i=0 U i =, we obtain TC() m+1 and hence we have TC() cat * (d()) + 1. Thus we have TC() = cat* (d()) + 1.

5 TOPOLOGICAL COMPLEXITY IS A FIREWISE L-S CATEGORY 5 3. Proof of Theorem 1.11 Assume that wgt (u; R) =m, where u H (, ()) and f : Y d() = a map of secat(f π) <m. Then there is an open cover S m i=1 U i = Y and a series of maps {σ i : U i P() ; 1 i m} satisfying π σ i = f Ui. Let Ŷ = Y q with projection pŷ and section sŷ given by pŷ Y = p Y, pŷ = id and sŷ : Y q = Ŷ. Then we can extend f to a map ˆf : Ŷ d() by the formula ˆf Y = f, ˆf = s d() =. y putting Ûi = U i q which is open in Ŷ, we obtain an open cover S m i=1 Ûi = Ŷ and a series of maps ˆσ i : Ûi P() satisfying π ˆσ i = ˆf Ûi : ˆσ i Ui = σ i, ˆσ i = s P(). Hence there is a fibrewise homotopy Φ i : Ûi [0, 1] d() such that Φ i (y, 0) = ˆf(y) and Φ i (y, 1) () given by the following formula. Φ i (y, t) = (ˆσ i (y)(t), ˆσ i (y)(1)), (y, t) Ûi [0, 1], so that we have Φ i (y, 0) = (ˆσ i (y)(0), ˆσ i (y)(1)) = π ˆσ i (y) = ˆf(y) and Φ i (y, 1) = (ˆσ i (y)(1), ˆσ i (y)(1)) (). Moreover, for any (b, t) [0, 1], we have Φ i (b, t) = (ˆσ i (b)(t), ˆσ i (b)(1)) = (s P() (t),s P() (1)) = (b, b). Thus Φ i gives a fibrewise pointed compression homotopy of ˆf Ûi into (). Then it follows that cat ( ˆf) <mand hence we obtain f (u) = 0 and wgt π (u; R) m. Thus we obtain wgt π (u; R) m = wgt (u; R). Conversely assume that wgt π (u; R) =m, where u H (, ()) and f : Y such that cat (f) <m. Then there exists an open covering S m i=1 U i = Y with U i s Y () and a sequence of fibrewise homotopies {φ i : U i [0, 1] } such that φ i (y, 0) = f Ui (y), φ i (y, 1) () and pr 2 φ i (y, t) = pr 2 f(y) for (y, t) U i [0, 1]. Hence there is a sequence of maps {σ i : U i P()} given by σ i (y)(t) = pr 1 φ i (y, t), y U i,t [0, 1] such that π σ i (y) = (pr 1 φ i (y, 0), pr 1 φ i (y, 1)) = f(y) since pr 2 φ i (y, t) = pr 2 f(y) for (y, t) U i [0, 1]. Thus we obtain secat(f π) <m, and hence f (u) = 0. This implies wgt (u; R) m = wgt π(u; R) and hence wgt (u; R) = wgt π(u; R). 4. Proof of Theorem 1.12 The proof of Lemma 2 in 2 of Milnor [16] implies the following: Lemma 4.1. The pair (, ()) is an NDR-pair. S Proof : For each vertex β of, let V β be the star neighbourhood in and V = β V β V β. Then the closure V = S V β β V β is a subcomplex of. For the barycentric coordinates {ξ β } and {η β } of x and y, resp, we see that (x, y) V if and only if P β Min(ξ β,η β ) > 0 and that P β Min(ξ β,η β ) = 1 if and only if the barycentric coordinates of x and y are the same, or equivalently, (x, y) (). Hence we can define a continuous map v : [ 1, 1] by the following formula. ( 2 P β v(x, y) = Min(ξ β,η β ) 1, if (x, y) V, 1, if (x, y) 6 V.

6 6 IWASE AND SAKAI Then we have that v 1 (1) = (). Let U = v 1 ((0, 1]) an open neighbourhood of (). Using Milnor s map s, we obtain a pair of maps (u, h) as follows: u(x, y) = Min{1, 1 v(x, y)} and h(x, y, t) = (s(x, y)(min{t, w(x, y)}),y), where w(x, y) =u(x, y) +v(x, y) = Min{1, 1+v(x, y)}. Note that w(x, y) = 1 if (x, y) U and that w(x, y) = 0 if (x, y) 6 V. Then u 1 (0) = (), u 1 ([0, 1)) = U and h(x, y, 1) = (y, y) () if (x, y) U. Moreover, pr 2 h(x, y, t) =y and h(x, x, t) = (s(x, x)(t),x) = (x, x) for any x, y and t [0, 1]. Thus the data (u, h) gives the fibrewise Strøm structure on (, ()). 5. Proof of Theorem 1.13 Let X be a fibrewise well-pointed space over and ˆX the fiberwise pointed space obtained from X by giving a fibrewise whisker. More precisely, we define ˆX be the mapping cylinder of s X, ˆX = X sx [0, 1], X 3 s X (b) (b, 0) [0, 1] for any b, with projection p ˆX and section s ˆX given by the formulas p ˆX X = p X, p ˆX [0,1] (b, t) =b, for (b, t) [0, 1], s ˆX(b) = (b, 1) [0, 1] ˆX. Then by the definition of Strøm structure, X is fibrewise pointed homotopy equivalent to ˆX the fibrewise whiskered space over. So we have cat (X) = cat ( ˆX) and cat * (X) = cat* ( ˆX). Assume that cat (X) =m 0. Then it is clear by definition that cat* (X) m = cat (X). S Conversely assume that cat * (X) = m 0. Then there is an open cover m i=0 U i = X such that U i is compressible into s X () X. Hence there is a fibrewise homotopy Φ i : U i [0, 1] X such that Φ i (x, 0) = x, Φ i (x, 1) = s X (p X (x)) and p X Φ i (x, t) =p X (x). We define Ûi as follows: Û i = U i sx (s X ) 1 (U i ) [0, 1] ( 2 3, 1]. We also define a fibrewise pointed homotopy ˆΦ i : Ûi [0, 1] ˆX as follows: Φ i (x, t), ˆx = x X, Φ i (s X (b),t 3s), ˆx =(b, s) (s X ) 1 (U i ) (0, t 3 ), s X (b), ˆx =(b, ˆΦ t i (ˆx, t) = 3 ),b (s X) 1 (U i ), (b, 6s 2t 6 3t ), ˆx =(b, s) (s X) 1 (U i ) ( t 3, 2 3 ), (b, 2 3 ), ˆx =(b, 2 3 ),b (s X) 1 (U i ), (b, s), ˆx =(b, s) ( 2 3, 1]. It is then easy to see that Ûi s cover the entire X, and hence we have cat ( ˆX) m = cat * (X). Thus cat (X) cat* (X) and hence cat (X) = cat* (X). In particular, we have TC() =TC M () for a locally finite simplicial complex.

7 TOPOLOGICAL COMPLEXITY IS A FIREWISE L-S CATEGORY 7 6. Fibrewise A structures From now on, we work in the category T. For any X a fibrewise pointed space over, we denote by p X : X its projection and by s X : X its section. We say that a pair (X, A) of fibrewise pointed spaces over is a fibrewise NDRpair or that A is a fibrewise NDR subset of X, if the inclusion map A X is a fibrewise cofibration, in other words, the inclusion has the fibrewise (strong) Strøm structure (see Crabb-James [1]). Since is the zero object in T, for any given fibrewise pointed space X over, we always have a pair (X, ) in T, where we regard s X () =. When the pair (X, ) is fibrewise NDR, the space X is called fibrewise well-pointed. Proposition 6.1 (Crabb-James [1]). (1) If (X, A) and (X 0,A 0 ) are fibrewise NDR-pairs, then so is (X, A) (X 0,A 0 ) = (X X 0,X A 0 A X 0 ). (2) If (X, A) is a fibrewise NDR-pair, then so is ( m Π X, m T (X, A)), which is defined by induction for all m 1: ( 1 Π X, 1 T (X, A)) = (X, A), ( m+1 Π X, m+1 T (X, A)) = ( m Π X, m T (X, A)) (X, A). If X is a fibrewise pointed space over, then by taking A =, we obtain a fibrewise subspace m+1 T (X, ) of m+1 T X, which is called an (m+1)-fold fibrewise fatwedge of X, and is often denoted by m+1 T X. In addition, the pair ( m+1 Π X, m+1 T X) is a fibrewise NDR-pair for all m 0, if X is fibrewise well-pointed. Examples 6.2. (1) Let X be a fibrewise pointed space over with p X = pr 2 : X = F the canonical projection to the second factor and s X = in 2 : F = E the canonical inclusion to the second factor. Then X is a fibrewise pointed space over. (2) Let X =, p X = pr 2 : the canonical projection to the second factor and s X = : the diagonal. Then X is a fibrewise pointed space over. (3) Let G be a topological group, EG the infinite join of G with right G action and G = EG/G the classifying space of G. y considering G as a left G space by the adjoint action, we obtain a fibrewise pointed space X = EG G G with p X : EG G G G with section s X : G EG G {e} EG G G. (4) Let be a space, X = L() the space of free loops on. Then p X : L() the evaluation map at 1 S 1 C is a fibration with section s X : L() given by the inclusion of constant loops. In view of Milnor s arguments, this example is homotopically equivalent to the example (3). Definition 6.3. Let P (X) = ` : [0, 1] X b s.t. t [0,1] p X (`(t))=b the fibrewise free path space, L (X) ={` P (X) `(1)=`(0)} the fibrewise free loop space and L (X) ={` P (X) `(1)=`(0)=s X p X (`(0))} the fibrewise pointed loop space. For any m 0, we define an A structure of L (X) as follows. (1) E m+1 (L (X)) as the homotopy pull-back in T m+1 of Π X - m+1 T X,

8 8 IWASE AND SAKAI (2) P m(l (X)) as the homotopy pull-back in T m+1 of X m+1 Π X - m+1 T X, (3) e X m : P m(l (X)) X as the induced map from the inclusion m+1 T X m+1 Π X by the diagonal m+1 : X m+1 Π X and (4) p L (X) : E m+1 (L (X)) P m(l (X)) as a map of fibrewise pointed spaces induced from the section s X : X, since the section m+1 Π X is nothing but the composition m+1 s X : s X m+1 m+1 Π X. We further investigate to understand an A stucture in a fiberwise view point, using fibrewise constructions. Clearly, these constructions are not exactly the Ganea-type fibre-cofibre constructions but the following. Proposition 6.4 (Sakai). Let X be a fibrewise pointed space over and m 0. Then P m+1 (L (X)) has the homotopy type of a push-out of p L (X) : E m+1 (L (X)) P m(l (X)) and the projection E m+1 (L (X)). This is a direct consequence of the following lemma. Lemma 6.5. Let (X, A) and (X 0,A 0 ) be fibrewise NDR-pairs of fibrewise pointed spaces over and Z a fibrewise pointed space over with fibrewise maps f : Z X and g : Z X 0. Then the homotopy pull-back Ω (f,g),k of maps (f, g) : Z X X 0 and k : X A 0 A X 0 X X 0 has naturally the homotopy type of the reduced homotopy push-out W =Ω g,j p2 Ω(f,g),i j ( J + ) p1 Ω f,i of p 1 :Ω (f,g),i j Ω f,i and p 2 :Ω (f,g),i j Ω g,j, where J =[ 1, 1] and Ω (f,g),k = n (z, `, `0) Z P (X) P (X 0 ) Ω (f,g),i j = (z, `, `0) Ω (f,g),k (`(1),`0(1)) A A 0, Ω f,i = {(z, `) Z P (X) f(z)=`(0), `(1) A}, Ω g,j = {(z, `0) Z P (X 0 ) g(z)=`0(0), `0(1) A 0 }, p 1 (z, `, `0) = (z, `) and p 2 (z, `, `0) = (z, `0). o f(z)=`(0), g(z)=`0(0),, (`(1),`0(1)) A X 0 X A 0 Proof of Outline of the proof. The proof of Lemma 6.5 is quite similar to that of Theorem 1.1 in Sakai [20] (which is based on Iwase [7]) by replacing (Y, ) in [20] by (X 0,A 0 ), defining and using the following spaces. cw =Ω (f,g),i idx 0 { 1} Ω (f,g),i j J Ω (f,g),idx j {1} Ω (f,g),k J, Ω (f,g),idx j = (z, `, `0) Ω (f,g),k (`(1),`0(1)) X A 0, Ω (f,g),i idx 0 = (z, `, `0) Ω (f,g),k (`(1),`0(1)) A X 0. The precise construction of homotopy equivalences and homotopies is identical to that in [20] and is left to the readers. Theorem 6.6. Let X be a fibrewise well-pointed space over. Then the sequence {p L (X) : E m+1 (L (X)) P m(l (X))} gives a fibrewise pointed version of A - structure on the fibrewise pointed loop space L (X). Thus in the case when X is a fibrewise well-pointed space over, we assume that P m (L (X)) is an increasing sequence given by homotopy push-outs with a

9 TOPOLOGICAL COMPLEXITY IS A FIREWISE L-S CATEGORY 9 fibrewise fibration e X m : P m(l (X)) X such that e X 1 fibrewise evaluation. : S (L (X)) X is a Examples 6.7. (1) Let X be a fibrewise pointed space over with p X = pr 2 : F the canonical projection and s X = in 2 : F the canonical inclusion. Then L (X) =L(F ) is given by p L (X) = pr 2 : L(F ) and s L (X) = in 2 : L(F ). (2) Let X = be a fibrewise pointed space over with p X = pr 2 : and s X = : the diagonal. Then L (X) =L() the free loop space on, p L (X) : L() the evalation map at 1 S 1 C and s L (X) : L() the inclusion of constant loops. Remark 6.8. When E is a cell-wise trivial fibration on a polyhedron (see [12]), we can see that the canonical map e E : P (L (E)) E is a homotopy equivalence by a similar arguments given in the proof of Theorem 2.9 of [12]. 7. Fibrewise L-S categories of fibrewise pointed spaces The fibrewise pointed L-S category of an fibrewise pointed space is first defined by James and Morris [13] as the least number (minus one) of open subsets which cover the given space and are contractible by a homotopy fixing the base point in each fibre (see also James [14] and Crabb-James [1]) and is redefined by Sakai in [19] as follows: let X be a fibrewise pointed space over. For given k 0, we denote by k+1 Π X the (k+1)-fold fibrewise product and by k+1 T X the (k+1)-fold fibrewise fat wedge. Then cat (X) m if the (m+1)-fold fibrewise diagonal map m+1 : X m+1 Π X is compressible into the fibrewise fat wedge m+1 T X in T. If there is no such m, we say cat (X) =. Let us consider the case when cat (X) <. The definition of a fibrewise A structure yields the following criterion. Theorem 7.1. Let X be a fibrewise pointed space over and m 0. cat (X) m if and only if id X : X X has a lift to P m (L (X)) ex m X in T. Proof : Then If cat (X) m, then the fibrewise diagonal m+1 : X m+1 Π X is compressible into the fibrewise fat wedge m+1 T X m+1 Π X in T. Hence there is a map σ : X P m(l (X)) in T such that ex m σ 1 X in T. The converse is clear by the definition of P m(l (X)). In the rest of this section, we work within the category T of fibrewise unpointed spaces and maps between them. ut we concentrate ourselves to consider its full subcategory T of all fibrewise pointed spaces, so in T, we have more maps than in T while we have just the same objects as in T. Let X be a fibrewise pointed space over. For given k 0, we denote by k+1 Π X the (k+1)-fold fibrewise product and by k+1 T X the (k+1)-fold fibrewise fat wedge. Then cat * (X) m if the (m+1)-fold fibrewise diagonal map m+1 : X m+1 Π X

10 10 IWASE AND SAKAI is compressible into the fibrewise fat wedge m+1 T X in T. If there is no such m, we say cat * (X) =. Let us consider the case when cat* (X) <. The definition of a fibrewise A structure yields the following. Theorem 7.2. Let X be a fibrewise pointed space over and m 0. Then cat * (X) m if and only if id X : X X has a lift to P m (L (X)) ex m X in the category T. Proof : If cat * (X) m, then the fibrewise diagonal m+1 : X m+1 Π X is compressible into the fibrewise fat wedge m+1 T X m+1 Π X in T. Hence there is a map σ : X P m(l (X)) in T such that ex m σ 1 X in T. The converse is clear by the definition of P m(l (X)). 8. Upper and lower estimates For X a fibrewise pointed space over, we define a fibrewise version of Ganea s strong L-S category (see Ganea [6]) of X as Cat (X) and also a fibrewise version of Fox s categorical length (see Fox [5] and Iwase [10]) of X as catlen (X). Definition 8.1. Let X be a fibrewise pointed space over. (1) Cat (X) is the least number m 0 such that there exists a sequence {(X i,h i ) h i : A i X i 1, 0 i m} of pairs of space and map satisfying X 0 = and X m ' X in T with the following homotopy push-out diagrams: A i p Ai h i X i 1 X i s Xi (2) catlen (X) is the least number m 0 such that there exists a sequence {X i h i : A i X i 1, 0 i m} of spaces satisfying X 0 = and X m ' X in T and that : X i X i X i is compressible into X i X i 1 X i in X m X m. A lower bound for the fibrewise L-S category of a fibrewise pointed space X over can be described by a variant of cup length: since X is a fibrewise pointed space over, there is a projection p X : X with its section s X : X. Hence we can easily observe for any multiplicative cohomology theory h that h (X) = h () h (X, ), where we may identify h (X, ) with the ideal ker s X : h (X) h (). Definition 8.2. For a fibrewise pointed space X over and any multiplicative cohomology theory h, we define cup (X; h) = Max {m 0 {u 1,,u m h (X, )} s.t. u 1 u m 6=0}, cup (X) = Max cup (X; h) h is a multiplicative cohomology theory.

11 TOPOLOGICAL COMPLEXITY IS A FIREWISE L-S CATEGORY 11 We often denote cup (;h) by cup (;R) when h ( ) = H (;R), where R is a ring with unit. Let us recall that the relationship between an A -structure and a Lusternik- Schnirelmann category gives the key observation in [7, 8, 9]. On the other hand, Rudyak [17] and Strom [23] introduced a homotopy theoretical version of Fadell-Husseini s category weight, which can be translated into our setting as follows: for any fibrewise pointed space X over, let {p L (X) k : E k (L (X)) P k 1 (L (X)) ; k 1} be the fibrewise A -structure of L (X) in the sense of Stasheff [22] (see also [11] for some more properties). Let h be a generalisd cohomology theory. Definition 8.3. For any u h (X, ), we define wgt (u; h) = Min m 0 Ø Ø (e X m) (u) 6= 0, where e X m is the composition of fibrewise maps P m(l (X)) P (L (X)) ex X. ' Using this, we introduce some more invariants as follows. Definition 8.4. For any fibrewise pointed space X over, we define wgt π (X; h) = Max {wgt π (u; h) u h (X, )}, wgt π (X) = Max {wgt π (X; h) h is a generalised cohomology theory}, wgt (X; h) = Max wgt (u; h) u h (X, ), wgt (X) = Max wgt (X; h) h is a generalised cohomology theory. We often denote wgt π (;h) and wgt (;h) by wgt π(;r) and wgt (;R) respectively when h ( ) = H (;R), where R is a ring with unit. We define versions of module weight for a fibrewise pointed space over. Definition 8.5. For a fibrewise pointed space X over, we define æ (1) Mwgt Ωm 0 (X; h) = Min Ø (ex m) is a split mono of (unstable) h h- for modules a generalisd cohomology theory h. (2) Mwgt (X) = Max Mwgt (X; h) h is a generalised cohomology theory. Then we immediately obtain the following result. Theorem 8.6. For any fibrewise pointed space X over, we have cup (X) wgt (X) Mwgt (X) cat (X) catlen (X) Cat (X). y Lemma 4.1, we have the following as a corollary of Theorem Corollary 8.7. For any space having the homotopy type of a locally finite simplicial complex, we obtain Z π () wgt π () Mwgt (d()) T C() 1 catlen (d()) Cat (d()).

12 12 IWASE AND SAKAI 9. Higher Hopf invariants For any fibrewise pointed map f : S (V ) X in T, we have its adjoint ad f : V L (X) such that e X 1 S (ad f) =f : S (V ) X. If cat (X) m, then there is a fibrewise pointed map σ : X P m L (X) in T such that e X 1 σ ' id X : X X. Hence both the fibrewise maps e X 1 (σ f) and e X 1 S (ad f) are fibrewise pointed homotopic to f in T. Then we have e X 1 {S (ad f) (σ f)} ', where ' denotes the fibrewise pointed homotopy and denotes the fibrewise trivial map in T. Thus there is a fibrewise pointed map Hσ m(f) : S (V ) E m+1 L (X) such that p L m (X) Hm(f) σ ' S (ad f) (σ f). Definition 9.1. Let X be of cat (X) m, m 0. For f : S (V ) X, we define (1) Hm(f) = Hm(f) σ e X 1 σ ' id X [S (V ),X], (2) Hm(f) = (S ) Hm(f) σ e X 1 σ ' id X S (V ),X, where, for two fibrewise spaces V and W, we denote by {V, W } the homotopy set of fibrewise stable maps from V to W. Appendix A. Fibrewise homotopy pull-backs and push-outs In this paper, we are using A structures which is constructed using tools in T and T especially, finite homotopy limits and colimits, in other words, fibrewise homotopy pull-backs and push-outs in T and T. We show in this section that such constructions are possible even when a fibrewise space has some singular fibres. First we consider the fibrewise homotopy pull-backs in T : let X, Y, Z and E be fibrewise spaces over and p : E Z be a fibrewise fibration in T. For any fibrewise map f : X Z in T, there exists a pull-back X X f Z p E as f E = {(x, e) X Ef(x) =p(e)} f p f E ˆf E of a subspace of X E together with fibrewise maps f p : f E X and ˆf : f E E given by restricting canonical projections: (f p)(x, e) =x, ˆf(x, e) =e. Theorem A.1 (Crabb-James [1]). Let p : E Z be a fibrewise fibration. For any fibrewise map f : W Z in T, f p : f E W is also a fibrewise fibration. Let π t : P (Z) Z be fibrewise fibrations given by π t (`) =`(t), t =0, 1 (see also [1]). Then π 0 and π 1 induce a map π : P (Z) Z Z to the fibre product of two copies of p Z : Z.

13 TOPOLOGICAL COMPLEXITY IS A FIREWISE L-S CATEGORY 13 Proposition A.2. π : P (Z) Z Z is a fibrewise fibration. Proof : For any fibrewise map φ : W P (Z) and a fibrewise homotopy H : W [0, 1] = W (I ) Z Z such that H(w, 0) = π φ(w) for w W, we define a fibrewise homotopy Ĥ : W [0, 1] = W (I ) P (Z)( P(Z)) by pr 0 H(w, s), if t = 0, pr 0 H(w, s 3t), if 0 < t < s 3, π 0 φ(w), if t = s 3, Ĥ(w, s)(t) = φ(w)( 3t s 3 2s ), if s 3 < t < 3 s 3, π 1 φ(w), if t = 3 s 3, pr 1 H(w, 3t 3+s), if 3 s 3 < t < 1 pr 1 H(w, s), if t = 0, for (w, s) W I and t [0, 1], where pr k : Z Z Z Z Z denotes the canonical projection given by pr k (z 0,z 1 )=z k, k =0, 1 for any (z 0,z 1 ) Z Z. Then for any (w, s) W I, we clearly have Ĥ(w, 0)(t) =φ(w)(t), t [0, 1], (Ĥ(w, s)(0), Ĥ(w, s)(1)) = (pr 0 H(w, s), pr 1 H(w, s)) = H(w, s), and hence we have Ĥ(w, 0) = φ(w) for any w W and also π Ĥ = H. This implies that Ĥ is a fibrewise homotopy of φ covering H. Thus π is a fibrewise fibration. This yields the following corollary. Corollary A.3. For any fibrewise maps f : X Z and g : Y Z in T, the induced map (f g) π :(f g) P (Z) X Y is a fibrewise fibration in T. We often call the fibrewise space (f g) P (Z) together with the projections pr X (f g) π :(f g) P (Z) X and pr Y (f g) π :(f g) P (Z) Y the homotopy pull-back in T of X f Z g Y. We remark that the above construction can be performed within T if X, Y, Z, f and g are all in T, so that we have a pointed version of a fibrewise homotopy pull-back: Corollary A.4. For any fibrewise maps f : X Z and g : Y Z in T, the induced map (f g) π :(f g) P (Z) X Y is a fibrewise fibration in T. Second we consider the fibrewise homotopy push-outs in T : let X, Y, Z and W be fibrewise pointed spaces over and i : Z W be a fibrewise cofibration in T. For any fibrewise map f : Z X over, there exists a push-out X f i f W ˇf W of X f Z i W as a quotient space of Xq W by gluing f(z) with i(z) together with fibrewise maps f i and ˇf induced from the canonical inclusions. Theorem A.5 (Crabb-James [1]). Let i : Z W be a fibrewise cofibration in T (or T ). For any fibrewise map f : Z X in T (or T, resp.), f i : X f W is also a fibrewise cofibration in T (or T, resp.). Let us recall that I (Z) is obtained from I (Z) =Z ( [0, 1]) = Z [0, 1] by identifying the subspace s Z () [0, 1] Z [0, 1] with s Z () by the canonical

14 14 IWASE AND SAKAI projection to the first factor : s Z () [0, 1] s Z (). Let ι t : Z I (Z) be fibrewise cofibration in T given by ι t(z) =q(z, t), 0 t 1, where q : Z [0, 1] I (Z) denotes the identification map. Then ι 0 and ι 1 induce a map ι : Z Z I (Z) from Z Z the push-out of two copies of s Z : Z. Proposition A.6. ι : Z Z I (Z) is a fibrewise cofibration. Proof : For any fibrewise map φ : I (Z) W and a fibrewise homotopy H : (Z Z) [0, 1] = (Z Z) I W such that H(z,0) = φ ι(z) for z Z Z, we define a fibrewise homotopy Ȟ : I (Z) [0, 1] = I (Z) (I ) W by H(in 0 (z),s 3t), if 0 t s 3, Ȟ(q(z, t), s)= φ(q(z, 3t s 3 2s )), if s 3 t 3 s 3, H(in 1 (z), 3t 3+s), if 3 s 3 t 1 for (q(z, t),s) I (Z) I, where in k : Z Z Z, k =0, 1 denote the canonical inclusion given by in 0 (z) = (z, b ) and in 1 (z) = ( b,z), b = p Z (z) for any z Z. Then for any (q(z, t),s) I (Z) I, we clearly have Ȟ(q(z, t))(0) = φ(q(z, t)), Ȟ(q(z,0))(s) =H(in 0 (z),s), Ȟ(q(z,1))(s) =H(in 1 (z),s), and hence we have Ȟ(q(z, t))(0) = φ(q(z, t)) for any q(z, t) I (Z) and also Ȟ (ι 1 I )=H. This implies that Ȟ is a fibrewise homotopy of φ extending H. Thus ι is a fibrewise cofibration. This yields the following corollary. Corollary A.7. For any fibrewise maps f : Z X and g : Z Y in T, the induced map (f g) ι : X Y (f g) I (Z) is a fibrewise cofibration in T. We often call the fibrewise space (f g) I (Z) together with the inclusions (f g) ι in X : X (f g) I (Z) and (f g) ι in Y : Y (f g) I (Z) as homotopy push-out in T of X f Z g Y. Quite similarly for a fibrewise space Z in T, we obtain a fibrewise cofibration ˆι : Z q Z = Z {0} Z {1} Z [0, 1] = I (Z). Thus we have the following. Corollary A.8. For any fibrewise maps f : Z X and g : Z Y in T, the induced map (fqg) ˆι : XqY (fqg) I (Z) is a fibrewise cofibration in T. Thus we also have an unpointed version of a fibrewise homotopy push-out. References [1] M. C. Crabb. and I. M. James, Fibrewise Homotopy Theory, Springer Monographs in Mathematics, Springer-Verlag London, Ltd., London, [2] M. Farber, Topological complexity of motion planning, Discrete Comput. Geom. 29 (2003), [3] M. Farber, Topology of robot motion planning, Morse theoretic methods in nonlinear analysis and in symplectic topology, , NATO Sci. Ser. II Math. Phys. Chem., 217, Springer, Dordrecht, [4] M. Farber and M. Grant, Symmetric Motion Planning, Topology and robotics, , Contemp. Math., 438, Amer. Math. Soc., Providence, RI, [5] R. H. Fox, On the Lusternik-Schnirelmann category, Ann. of Math. (2) 42, (1941),

15 TOPOLOGICAL COMPLEXITY IS A FIREWISE L-S CATEGORY 15 [6] T. Ganea, Lusternik-Schnirelmann category and strong category, Illinois. J. Math. 11 (1967), [7] N. Iwase, Ganea s conjecture on Lusternik-Schnirelmann category, ull. Lon. Math. Soc., 30 (1998), [8] N. Iwase, A -method in Lusternik-Schnirelmann category, Topology 41 (2002), [9] N. Iwase, Lusternik-Schnirelmann category of a sphere-bundle over a sphere, Topology 42 (2003), [10] N. Iwase, Categorical length, relative L-S category and higher Hopf invariants, preprint. [11] N. Iwase and M. Mimura, Higher homotopy associativity, Algebraic Topology, (Arcata CA 1986), Lect. Notes in Math. 1370, Springer Verlag, erlin (1989) [12] N. Iwase and M. Sakai, Functors on the category of quasi-fibrations, Topology Appl., to appear. [13] I. M. James and J. R. Morris, Fibrewise category, Proc. Roy. Soc. Edinburgh. 119A (1991), [14] I. M. James, Introduction to fibrewise homotopy theory, Handbook of algebraic topology, , North Holland, Amsterdam, [15] I. M. James, Lusternik-Schnirelmann Category, Handbook of algebraic topology, , North Holland, Amsterdam, [16] J. Milnor, On Spaces Having the Homotopy Type of a CW-Complex, Trans. Amer. Math. Soc. 90 (1959), [17] Y.. Rudyak, On category weight and its applications, Topology 38 (1999), [18] Y.. Rudyak, On analytical applications of stable homotopy (the Arnold conjecture, critical points), Math. Z. 230(1999) [19] M. Sakai, The functor on the category of quasi-fibrations, DSc Thesis (Kyushu University 1999), [20] M. Sakai, A proof of the homotopy push-out and pull-back lemma, Proc. Amer. Math. Soc. 129 (2001), [21] A. S. Schwarz The genus of a fiber space, Amer. Math. Soc. Transl.(2) 55 (1966), [22] J. D. Stasheff, Homotopy associativity of H-spaces, I, II, Trans. Amer. Math. Soc. 108 (1963), , [23] J. Strom, Essential category weight and phantom maps, Cohomological methods in homotopy theory (ellaterra, 1998), , Progr. Math., 196, irkhauser, asel, [24] G. W. Whitehead, Elements of Homotopy Theory, Springer Verlag, erlin, GTM series 61, address: iwase@math.kyushu-u.ac.jp address, Sakai: sakai@gifu-nct.ac.jp (Iwase) Faculty of Mathematics, Kyushu University, Fukuoka , Japan (Sakai) Gifu National College of Technology, Gifu , Japan.