Robot motion planning, weights of cohomology classes, and cohomology operations

The complexity of algorithms solving the motion planning problem is measured by a homotopy invariant TC(X) of the configuration space X of the system. Previously known lower bounds for TC(X) use the structure of the cohomology algebra of X. In this paper we show how cohomology operations can be used to sharpen these lower bounds for TC(X). As an application of this technique we calculate explicitly the topological complexity of various lens spaces. The results of the paper were inspired by the work of E. Fadell and S. Husseini on weights of cohomology classes appearing in the classical lower bounds for the Lusternik - Schnirelmann category. In the appendix to this paper we give a very short proof of a generalized version of their result.


Introduction
The motion planning problem is a central theme of robotics. Given a mechanical system S, a motion planning algorithm for S is a function which associates to any pair of states (A, B) of S a continuous motion of the system starting at A and ending at B. If X denotes the configuration space of the system, one considers the path fibration π : X I → X × X, π(γ) = (γ(0), γ(1)), γ : I → X, (1) and, in these terms, a motion planning algorithm for S is a section (not necessarily continuous) of π. The topological complexity of X, denoted TC(X), is defined to be the genus, in the sense of Schwarz, of fibration (1). The concept TC(X) was introduced and studied in [2], [4]; it is a measure of complexity of the problem of finding a motion planning algorithm for a system whose configuration space is homotopy equivalent to X. A recent survey of related results can be found in [5].
A lower bound for TC(X) was given in [2] in terms of zero-divisors in cohomology with coefficients in a field k. The cup product map is an algebra homomorphism, whose kernel is called the ideal of zero-divisors. The multiplicative structure on the left in (2) is given by the formula (α ⊗ β)(γ ⊗ δ) = (−1) |β||γ| αγ ⊗ βδ. A theorem from [2] claims that TC(X) is greater than the zerodivisors cup-length of X, where the latter is defined as the length of the longest non-trivial product of elements in the ideal of zero-divisors. This lower bound is sharp in many cases, and is easy to apply as it only requires knowledge of the cohomology algebra of X.
In this paper we show how the lower bound for TC(X) mentioned above may be improved upon using cohomology operations. We employ the notion of weight of a cohomology class with respect to a fibration, introduced in [6], which generalises the notion of category weight developed by Fadell and Husseini [1] for estimating the Lusternik-Schnirelmann category of a space.
The main result of this paper is described in §3. Here we introduce the notion of excess of a stable cohomology operation. With the aid of this notion we find indecomposable zero-divisors having weight at least 2 with respect to the path fibration (1). This result is applied in §5 to the computation of the topological complexity of some lens spaces. In section §4 we give an upper bound for the topological complexity of a fibration which is used in §5.
Fadell and Husseini showed that certain cohomology classes which are the images of Steenrod operations have weight at least 2. As Appendix A to this paper we give a short proof of a more general result. Our Theorem 18 is more general than the result of [1] in two respects: we allow more general cohomology operations, and give estimates for the strict category weight of Rudyak [11] as opposed to the usual category weight of Fadell and Husseini [1].
All topological spaces are assumed to be path-connected, and all maps are continuous. All rings are assumed to be commutative and possess a unit.

Weight of a cohomology class with respect to a fibration
We start by recalling a result from [6] which will be used in this paper. Let p : E → B be a fibration. The Schwarz genus of p, denoted genus(p), is defined to be the least integer k such that the base B may be covered by k open sets U 1 , . . . , U k on each of which p admits a local section (a continuous map s i : U i → E satisfying p • s i = 1 Ui for i = 1, . . . , k). The concept of genus of a fibration was introduced and thoroughly studied by A. S. Schwarz [12]. In the literature the term sectional category is also used.
There are two important special cases when one is motivated to study the Schwarz genus. Let X be a topological space. Consider the Serre path fibration π 0 : P 0 X → X where the total space P 0 X consists of all paths γ in X with γ(0) equal to a fixed base point x 0 ∈ X. The projection π 0 takes a path γ to γ(1) ∈ X. The genus of this fibration equals the Lusternik -Schnirelmann category cat(X) = genus(π 0 ). Another special case is used to estimate complexity of the robot motion planning problem. Let X be a topological space and let X I be the space of all paths in X equipped with the compact-open topology. The map π : X I → X × X associates to a path γ ∈ X I the pair of its end points (γ(0), γ(1)). It is a fibration and its genus is called the topological complexity of X, denoted TC(X) = genus(π).
One thinks of X as being the configuration space of a mechanical system; then TC(X) measures the "navigational complexity" of X; see [5] for more detail.
A well-known lower bound for the genus of a fibration p : E → B is given by the cup-length of the kernel of the induced map p * in cohomology. Taking coefficients in an arbitrary ring R, suppose there are classes u 1 , . . . , u ℓ ∈ H * (B; R) with such that their cup product u 1 · · · u ℓ = 0 ∈ H * (B; R) is non-zero. Then one has genus(p) > ℓ, see [12], Theorem 4.
The notion of category weight of E. Fadell and S. Husseini [1] generalises to an arbitrary fibration p : E → B as follows, see [6]. Definition 1. Let u ∈ H * (B; G) be a cohomology class, where G is an abelian group. The weight of u with respect to p : E → B, denoted wgt p (u), is defined to be the largest integer k such that f * (u) = 0 ∈ H * (Y ; G) for all maps f : Y → B with genus(f * p) ≤ k.
Here f * p : E ′ → Y denotes the pull-back fibration of p along f and the inequality genus(f * p) ≤ k means that there exists an open cover U 1 ∪ · · · ∪ U k = Y and continuous maps φ i : Clearly wgt p (u) ≥ 0 for all classes u, and wgt p (u) ≥ 1 if and only if p * (u) = 0. It is convenient to define the weight of the zero cohomology class as being +∞.
For a proof see [6]. The lower bound for genus(p) given by this Proposition may improve upon that given by the cup-length of ker p * , if we can find indecomposables u ∈ H * (B) with wgt p (u) ≥ 2. In the next section we show how one may find such cohomology classes using cohomology operations.

TC-weights of cohomology classes and cohomology operations
In this section we study weights of cohomology classes in the context of topological complexity.
Definition 3. The TC-weight of a cohomology class u ∈ H * (X × X; G) is defined as its weight wgt π (u), in the sense of Definition 1, with respect to the path fibration π : X I → X × X.
As in general, wgt π (u) ≥ 1 if and only if π * (u) = 0. The latter condition can be replaced by ∆ * (u) = 0 where ∆ : X → X × X is the diagonal. If the group of coefficients G is a field then u can be viewed as an element of H * (X; G)⊗ H * (X; G) and the property ∆ * (u) = 0 can be expressed by saying that u is a zero-divisor in the sense of [2].
To find classes with wgt π (u) ≥ 2 we will need the following Lemma. Proof. We have a pull-back diagram π ?
The conclusion of the lemma follows immediately from the the following statement: There exists a local section of f * π over an open subset A ⊆ Y if and only if ϕ| A ≃ ψ| A . We first remark that a local section s A : Then we may define a homotopy F : Conversely, suppose we have a homotopy G : A × I → X from ϕ| A to ψ| A . Then we may define our map S A : A → X I by the formula We now describe a method for finding indecomposable classes with TC-weight more than one, using cohomology operations.
Let R and S be abelian groups. A stable cohomology operation of degree i is a family of natural transformations θ : H n (−; R) → H n+i (−; S), one for each n ∈ Z, which commute with the suspension isomorphisms, see [10]. It follows that θ commutes with all Mayer-Vietoris connecting homomorphisms, and each homomorphism (4) is additive, i.e. is a group homomorphism.
Definition 5. The excess of a stable cohomology operation θ, denoted e(θ), is defined to be the largest integer n such that θ(u) = 0 for all cohomology classes u ∈ H m (X; R) with m < n.
Consider a few examples. For any extension 0 → R ′ → R → R ′′ → 0 of abelian groups the Bockstein homomorphism has excess one. The excess of the Steenrod square equals i and for any odd prime p the excess of the Steenrod power operation equals 2i, see [9], pages 489 -490. More generally, the excess of a composition of It is easy to see that for an admissible sequence I = i 1 i 2 . . . i n (i.e. such that i k ≥ 2 · i k+1 for all k) the excess equals which coincides with the standard notion of excess, see [10], page 27.
Any cohomology class u ∈ H j (X; R) determines a class where × denotes the cohomology cross product. Note that u is a zero-divisor and hence wgt π (u) ≥ 1. Observe that , by the naturality and additivity of θ (here p 1 , p 2 : X × X → X are the projections onto each factor).
Note that similar results holds in a more general situation when θ : E * → F * +i is a stable cohomology operation between extraordinary cohomology theories.

Motion planning in fibre spaces
In this section we give an upper bound for the topological complexity of fibre spaces in terms of invariants of the base and fibre. It will be used in the following section in the study of lens spaces. Lemma 7. Let p : E → B be a Hurewicz fibration with fibre F . Then Proof. Denote k = cat(B × B) and ℓ = TC(F ). Suppose that are open covers such that each inclusion U i → B × B is null-homotopic and there exists a continuous section s i : V i → F I of the end-point map F I → F × F for each i = 1, . . . , ℓ. Fix a homotopy h j : U j → (B × B) I = B I × B I connecting the inclusion U j → B × B with the constant map onto (x 0 , x 0 ). For (x, y) ∈ U j the image h j (x, y) is a pair of paths (α x,y , β x,y ) in B satisfying α x,y (0) = x, α x,y (1) = x 0 and β x,y (0) = y, β x,y (1) = x 0 .
It is clear that the family {W j,i } is an open cover of E × E and over each set W j,i there is a continuous section of E I → E × E: if (e, e ′ ) ∈ W j,i then the connecting path is concatenation of λ(e, α x,y ), path s i (a, b) in the fibre F connecting a to b, and the reverse path to λ(e ′ , β x,y ). Hence, TC(E) ≤ kℓ.
We mention the following special cases: Corollary 8. Let E be the total space of a fibration with fibre F such that the base B is homotopy equivalent to a sphere S k . Then TC(E) ≤ 3 · TC(F ).

Topological complexity of lens spaces
In this section we apply Theorem 6 to the problem of computing topological complexity of lens spaces. Let m ≥ 2 be an integer. We regard the cyclic group Z m as the multiplicative group {1, ω, . . . , ω m−1 } ⊆ C of m-th roots of unity. This acts freely on the unit sphere S 2n+1 ⊆ C n+1 by pointwise multiplication. The quotient is the lens space L 2n+1 m = S 2n+1 /Z m . In the literature this space is known as L m (1, 1, . . . , 1), see page 144 of [9].
We start by improving a general upper bound TC(X) ≤ 2 · dim(X) + 1, see [2], for the lens spaces. There are many known examples when TC(X) = 2 dim(X) + 1; however, for real projective spaces one has a better upper bound TC(RP n ) ≤ 2n which is an equality if and only if n is a power of two, see [3], [5]. ; Z m ) is Z m for 0 ≤ i ≤ 2n + 1 and vanishes for i > 2n + 1, see [9]. As generators one can choose x ∈ H 1 (L 2n+1 m ; Z m ) and ; Z m ) viewed as a free Z m -module. Since β is a stable cohomology operation of excess 1, we have by Theorem 6 wgt π (β(x))) = wgt π (y) ≥ 2, where y = 1 ⊗ y − y ⊗ 1 ∈ H 2 (L 2n+1 m × L 2n+1 m ; Z m ) is a zero-divisor. If for some 0 ≤ k, ℓ ≤ n the binomial coefficient k+ℓ k is not divisible by m then the power (y) k+ℓ is nonzero since it contains the term (−1) k k+ℓ k y k ⊗y ℓ . The productx·(ȳ) k+l is also nonzero (for obvious reasons) and hence applying Proposition 2, we obtain TC(L 2n+1 m ) ≥ 2(k + ℓ) + 2. This completes the proof.
To state the following result we need a new notation. For an integer n we will denote by α(n) = α 2 (n) the number of ones in the dyadic expansion of n. To define a similar number α p (n), for any odd prime p, consider the p-adic representation of n, n = n 0 + n 1 p + · · · + n k p k , n i ∈ {0, 1, . . . , p − 1}. The number α p (n) is defined by counting indices i such that 2n i ≥ p, but our counting involves certain multiplicities r i = r i (n). We set r i = 0 iff 2n i < p. If 2n i ≥ p, we denote by r i ≥ 1 the maximal r ≥ 1 such that n i+1 = n i+2 = · · · = n i+r−1 = (p − 1)/2. Thus r i = 1 iff 2n i ≥ p and n i+1 = (p − 1)/2. Finally we define Examples: α 3 (13) = 0, α 3 (14) = 3.
assuming that m is divisible by p αp(n)+1 , for some prime p.
Proof. By Lemma 19 in Appendix B, the maximal power of p dividing 2n n is p αp(n) . Hence, the assumption of Theorem 12 can be equivalently expressed by saying that m does not divide 2n n . The result follows by combining the upper bound of Corollary 10 with the lower bound given by Theorem 11 (where we take k = ℓ = n).
The following statement is a useful special case of the previous theorem: Theorem 13. Suppose that p is an odd prime and n is such that its p-adic expansion, n = n 0 + n 1 · p + · · · + n k · p k , where n i ∈ {0, 1, . . . , m − 1}, involves only "digits" n i satisfying n i ≤ (p − 1)/2. Then the topological complexity of the (2n + 1)-dimensional lens space L 2n+1 Recall that α(n) denotes the number of ones in the dyadic expansion of n. We see that the topological complexity of lens spaces L 2n+1 4 equals twice the dimension for all n which are powers of 2. The topological complexity of lens spaces L 2n+1 8 equals twice the dimension for all n having at most two ones in their dyadic expansion; in other words, n must be the sum of at most two powers of two. ) obtained using Brown-Peterson cohomology. Papers [7], [8] contain also a general discussion comparing the problem of computing the topological complexity of lens spaces and the immersion problem for lens spaces, inspired by the result of [3].

Appendix A: Category weight of Fadell and Huseini
In this appendix we give a short proof of a result in the spirit of theorems of Fadell and Husseini [1]. Our Theorem 18 is slightly stronger than [1] since it gives lower bounds for the strict category weight of Y. Rudyak [11] instead of the original category weight of [1].
Definition 17 (Rudyak, [11]). Let u ∈ H * (X) be a cohomology class. The strict category weight of u, denoted swgt(u), is defined to be the largest integer k such that f * (u) = 0 for all maps f : Y → X with cat(f ) ≤ k. Recall that cat(f ) ≤ k means that Y may be covered by open sets U 1 , . . . , U k , the restriction of f to each of which is null-homotopic.
Clearly swgt(u) coincides with the weight wgt π0 (u) of u with respect to the Serre fibration π 0 : P 0 (X) → X. One may improve on the classical cup-length lower bound by finding indecomposable cohomology classes of category weight more than one. The following result includes Theorem 3.12 of [1] as a special case, see also Corollary 4.7 of [11].

Appendix B: Divisibility of binomial coefficients
For convenience of the reader we include the following well-known result: Lemma 19. Let p be a prime and n = n 0 + n 1 p + n 2 p 2 + . . . and m = m 0 + m 1 p + m 2 p 2 + . . . be p-adic representations of integers n and m, where 0 ≤ n i , m i < p. The maximal integer ℓ such that p ℓ divides the binomial coefficient n+m n equals the number of indices i = 0, 1, 2, . . . such that either n i + m i ≥ p (18) or, for some r ≥ 1, one has n i + m i = n i−1 + m i−1 = · · · = n i−r + m i−r = p − 1, n i−r−1 + m i−r−1 ≥ p. Proof. One observes that n! = p ℓ · n ′ where n ′ is an integer mutually prime to p and ℓ = i≥1