Topological complexity of the Klein bottle

Cohen, Daniel C.; Vandembroucq, Lucile

doi:10.1007/s41468-017-0002-0

Cited by 30 publications

(57 citation statements)

References 12 publications

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“…Since RP 2 admits an immersion into R 3 , the topological complexity of RP 2 is 3, see [11]. The topological complexity of all other non-orientable surfaces is 4 [3] (for surfaces of genus > 3 see [7]).…”

Section: Examplesmentioning

confidence: 99%

On the LS-category and topological complexity of a connected sum

Dranishnikov

Sadykov

2019

Proc. Amer. Math. Soc.

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The Lusternik-Schnirelmann category and topological complexity are important invariants of manifolds (and more generally, topological spaces). We study the behavior of these invariants under the operation of taking the connected sum of manifolds. We give a complete answer for the LS-categoryof orientable manifolds, cat (M #N ) = max{cat M, cat N }. For topological complexity we prove the inequality TC (M #N ) ≥ max{TC M, TC N } for simply connected manifolds.

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Section: Examplesmentioning

confidence: 99%

On the LS-category and topological complexity of a connected sum

Dranishnikov

Sadykov

2019

Proc. Amer. Math. Soc.

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“…In a recent breakthrough, the topological complexity of closed non-orientable surfaces of genus g ≥ 2 was computed by A. Dranishnikov for g ≥ 4 in [9] and by D. Cohen and L. Vandembroucq for g = 2, 3 in [5]. In both these articles it is shown that TC(π) attains its maximum, i.e.…”

Section: Introductionmentioning

confidence: 99%

Bredon cohomology and robot motion planning

Farber

Grant

Lupton

et al. 2019

Algebr. Geom. Topol.

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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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“…As a recent breakthrough, the topological complexity of closed non-orientable surfaces of genus g ≥ 2 has only recently been computed by A. Dranishnikov for g ≥ 4 in [11] and by D. Cohen and L. Vandembroucq for g = 2, 3 in [6]. In both these articles it is shown that TC(K(π, 1)) attains its maximum, i.e.…”

Section: Introductionmentioning

confidence: 99%

On the topological complexity of aspherical spaces

Farber

Mescher

2018

J. Topol. Anal.

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The well-known theorem of Eilenberg and Ganea [Ann. Math. 65 (1957) 517–518] expresses the Lusternik–Schnirelmann category of an Eilenberg–MacLane space [Formula: see text] as the cohomological dimension of the group [Formula: see text]. In this paper, we study a similar problem of determining algebraically the topological complexity of the Eilenberg–MacLane spaces [Formula: see text]. One of our main results states that in the case when the group [Formula: see text] is hyperbolic in the sense of Gromov, the topological complexity [Formula: see text] either equals or is by one larger than the cohomological dimension of [Formula: see text]. We approach the problem by studying essential cohomology classes, i.e. classes which can be obtained from the powers of the canonical class (as defined by Costa and Farber) via coefficient homomorphisms. We describe a spectral sequence which allows to specify a full set of obstructions for a cohomology class to be essential. In the case of a hyperbolic group, we establish a vanishing property of this spectral sequence which leads to the main result.

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Topological complexity of the Klein bottle

Abstract: We show that the normalized topological complexity of the Klein bottle is equal to 4. For any non-orientable surface N g of genus g ! 2, we also show that TCðN g Þ ¼ 4. This completes recent work of Dranishnikov on the topological complexity of non-orientable surfaces.

Cited by 30 publications

References 12 publications

On the LS-category and topological complexity of a connected sum

On the LS-category and topological complexity of a connected sum

Bredon cohomology and robot motion planning

On the topological complexity of aspherical spaces

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