A mapping theorem for topological complexity

In this paper we study the topological invariant TC(X) reflecting the complexity of algorithms for autonomous robot motion. Here, X stands for the configuration space of a system and TC(X) is, roughly, the minimal number of continuous rules which are needed to construct a motion planning algorithm in X. We focus on the case when the space X is aspherical; then the number TC(X) depends only on the fundamental group π = π1(X) and we denote it TC(π). We prove that TC(π) can be characterised as the smallest integer k such that the canonical π × π-equivariant map of classifying spacescan be equivariantly deformed into the k-dimensional skeleton of ED(π × π). The symbol E(π×π) denotes the classifying space for free actions and ED(π×π) denotes the classifying space for actions with isotropy in a certain family D of subgroups of π × π. Using this result we show how one can estimate TC(π) in terms of the equivariant Bredon cohomology theory. We prove that TC(π) ≤ max{3, cdD(π × π)}, where cdD(π × π) denotes the cohomological dimension of π × π with respect to the family of subgroups D. We also introduce a Bredon cohomology refinement of the canonical class and prove its universality. Finally we show that for a large class of principal groups (which includes all torsion free hyperbolic groups as well as all torsion free nilpotent groups) the essential cohomology classes in the sense of Farber and Mescher [17] are exactly the classes having Bredon cohomology extensions with respect to the family D. United Kingdom

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“…On the other hand the set X H is the union of the sets X j which are single points. Hence (23) is an isomorphism.…”

Section: Principal O D -Modulesmentioning

confidence: 91%

Bredon cohomology and robot motion planning

Farber

Grant

Lupton

et al. 2019

Algebr. Geom. Topol.

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“…We will adopt the non-standard notation Remark 5.4. One could also prove Theorem 1.1 by analysing the long exact sequence in homotopy of the fibration q : E → Y A × Y B (compare [21]). When the condition AB = G does not hold, the total space E is disconnected.…”

Section: Higman's Groupmentioning

confidence: 99%

“…In the special case when either A or B is normal in G, then G is a semi-direct product. These results should be compared with results in our companion paper [21], in which we treat non-aspherical spaces.…”

Section: Introductionmentioning

confidence: 95%

New lower bounds for the topological complexity of aspherical spaces

Grant

Lupton

Oprea

2015

Topology and its Applications

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Abstract. We show that the topological complexity of an aspherical space X is bounded below by the cohomological dimension of the direct product A × B, whenever A and B are subgroups of π 1 (X) whose conjugates intersect trivially. For instance, this assumption is satisfied whenever A and B are complementary subgroups of π 1 (X). This gives computable lower bounds for the topological complexity of many groups of interest (including semidirect products, pure braid groups, certain link groups, and Higman's acyclic four-generator group), which in some cases improve upon the standard lower bounds in terms of zerodivisors cup-length. Our results illustrate an intimate relationship between the topological complexity of an aspherical space and the subgroup structure of its fundamental group.

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“…Since the mapsh n,m+1 andψ θ are of degree 1, (18) implies that the above composition is homotopic to the right-hand term in (21). The equivalence in (22) is obtained in the same manner, now using the (surjective) map h n+1,m : ∆ n × ∆ m × I → ∆ n+1 × ∆ m given by h n+1,m ((t, s, u)) = ((1 − u)t, u, s) and its induced maph n+1,m which is of degree (−1) m . Proposition 7.3, the triangle in (2), and the functoriality of standard difference pinch maps yields:…”

Section: ) With These Maps the Following Diagram Is Commutativementioning

confidence: 99%

“…on applying the result of [5,Theorem 3.6] or [22,Corollary 2.9]. (Alternatively, the inequality TC (X) ≥ 4 follows directly from a zero-divisors calculation in mod 2 cohomology.)…”

Section: Application: Ganea's Condition For Topological Complexitymentioning

confidence: 99%

Hopf invariants for sectional category with applications to topological robotics

González

Grant

Vandembroucq

2019

The Quarterly Journal of Mathematics

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We develop a theory of generalized Hopf invariants in the setting of sectional category. In particular we show how Hopf invariants for a product of fibrations can be identified as shuffle joins of Hopf invariants for the factors. Our results are applied to the study of Farber's topological complexity for 2-cell complexes, as well as to the construction of a counterexample to the analogue for topological complexity of Ganea's conjecture on Lusternik-Schnirelmann category.MSC 2010: 55M30, 55Q25, 55S36, 68T40, 70B15.Example 1.3. If p is odd, 2p − 1 < q ≤ 3p − 3, and the join square H 0 (α) ⊛ H 0 (α) is a non-trivial element of odd torsion, then TC (X) = 4.Remark 1.4. We also get a full description of TC(X) for X = S p ∪ α e 2p (Theorems 5.1 and 5.2 below). The proofs are, however, much more elementary than those in the cases of Theorems 1.1 and 1.2.1 Base points will generically be denoted by an asterisk * . 2 These are not actual restrictions in view of [39, Theorem 5.95]. H n φ,α (p)⊛H m ψ,β (q) / / J n (A) ⊛ J m (B)

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A mapping theorem for topological complexity

Cited by 8 publications

References 30 publications

Bredon cohomology and robot motion planning

Bredon cohomology and robot motion planning

New lower bounds for the topological complexity of aspherical spaces

Hopf invariants for sectional category with applications to topological robotics

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