Computations in rational sectional category

Carrasquel-Vera, José Gabriel

doi:10.36045/bbms/1442364592

Cited by 8 publications

(13 citation statements)

References 10 publications

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“…By Hattori's work [14], complements of generic complex hyperplane arrangements are up-tohomotopy examples of the spaces dealt with in Theorem 1.3 (with k i = 1 for all i). Those spaces are known to be formal, so their rational higher topological complexity has been shown in [3] to agree with the cohomological lower bound. Of course, such an observation can be recovered from Theorem 1.3 in view of the general fact that the rational topological complexity bounds from below the regular one.…”

Section: Introductionmentioning

confidence: 94%

Higher topological complexity of subcomplexes of products of spheres and related polyhedral product spaces

González¹,

Gutiérrez²,

Yuzvinsky³

2016

TMNA

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We construct "higher" motion planners for automated systems whose space of states are homotopy equivalent to a polyhedral product space Z(K, {(S ki , ⋆)}), e.g. robot arms with restrictions on the possible combinations of simultaneously moving nodes. Our construction is shown to be optimal by explicit cohomology calculations. The higher topological complexity of other families of polyhedral product spaces is also determined.

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Section: Introductionmentioning

confidence: 94%

Higher topological complexity of subcomplexes of products of spheres and related polyhedral product spaces

González¹,

Gutiérrez²,

Yuzvinsky³

2016

TMNA

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“…Earlier versions of this work used the following generalization of the relations in (8): (3). Assume the result is valid for a fixed i with 1 ≤ i < k, and assume in addition (i + 1) + j + ℓ > m + k, then…”

Section: Cohomology and Ls-category Of F (1 K M)mentioning

confidence: 99%

Motion planning in real flag manifolds

González

Gutiérrez

et al. 2016

Homology, Homotopy and Applications

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Starting from Borel's description of the mod-2 cohomology of real flag manifolds, we give a minimal presentation of the cohomology ring for semi complete flag manifolds F k,m := F (1, . . . , 1, m) where 1 is repeated k times. The information is used in order to estimate Farber's topological complexity of these spaces when m approaches (from below) a 2-power. In particular, we get almost sharp estimates for F 2,2 e −1 which resemble the known situation for the real projective spaces F 1,2 e . Our results indicate that the agreement between the topological complexity and the immersion dimension of real projective spaces no longer holds for other flag manifolds. More interestingly, we also get corresponding results for the s-th (higher) topological complexity of these spaces. Actually, we prove the surprising fact that, as s increases, the estimates become stronger. Indeed, we get several full computations of the higher motion planning problem of these manifolds. This property is also shown to hold for surfaces: we get a complete computation of the higher topological complexity of all closed surfaces (orientable or not). A homotopy-obstruction-theory explanation is included for the phenomenon of having a cohomologically accessible higher topological complexity even when the regular topological complexity is not so accessible.

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“…We will start with a brief recall of some content of [2] and [1] that will be used later on. Throughout this paper we will work with commutative differential graded algebras over Q whose differential increases the degree.…”

Section: Module Sectional Category and Productsmentioning

confidence: 99%

“…By [2], one has that msecat(f ) = msecat(ϕ), for any surjective model ϕ of f . Recall that the nilpotency of an ideal I is defined as the greatest integer m such that I m+1 = {0}.…”

Section: This Induces Commutative Diagrammentioning

confidence: 99%