Lusternik–Schnirelmann category, complements of skeleta and a theorem of Dranishnikov

Oprea, John; Strom, Jeff

doi:10.2140/agt.2010.10.1165

Cited by 6 publications

(5 citation statements)

References 23 publications

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“…The Lusternik-Schnirelmann one-category, cat 1 (X), is defined as the sectional category secat(P ) of P . This interpretation of one-category goes back to Schwarz ([26]) who also showed that cat 1 (X) = cat(f), where f : X → K(π 1 (X), 1) classifies the universal cover and cat(f) is the category of the map f (also see [23]). This latter description easily implies, for instance, that cat 1 (T n × Y) = n when Y is simply connected.…”

Section: The D-topological Complexitymentioning

confidence: 84%

An upper bound for topological complexity

Farber

Grant

Lupton

et al. 2019

Topology and its Applications

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In [11], a new approximating invariant TC D for topological complexity was introduced called D-topological complexity. In this paper, we explore more fully the properties of TC D and the connections between TC D and invariants of Lusternik-Schnirelmann type. We also introduce a new TC-type invariant TC that can be used to give an upper bound for TC,where X is a finite dimensional simplicial complex with k-connected universal coverX. The above inequality is a refinement of an estimate given by Dranishnikov [5].

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Section: The D-topological Complexitymentioning

confidence: 84%

An upper bound for topological complexity

Farber

Grant

Lupton

et al. 2019

Topology and its Applications

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“…Proof. For k = 1, this is [OS,Theorem 6.1]. For k > 1, the proof is literally the same as for k = 1.…”

Section: Proofsmentioning

confidence: 89%

“…Now we prove the first part of 1.6. The proof is based (speculated) on [OS,Sections 5,6] that, in turn, exploits clever ideas of Dranishnikov [D1, D2].…”

Section: Proofsmentioning

confidence: 99%

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Cohomological Dimension, Connectivity, and Lusternik--Schnirelmann category

Rudyak

2017

Preprint

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where cd(π) denotes the cohomological dimension of a group π and hd(X) denotes the homotopy dimension of X. Furthermore, there is a well-known inequality of Grossman, [G]:We make a synthesis and generalization of both of these results, by demonstrating the main result:The proof of the main theorem uses the Oprea-Strom inequality cat X ≤ hd(Bπ 1 (X))+cat 1 X, [OS] where cat 1 is the Clapp-Puppe cat A with A the class of 1-dimensional CW complexes. The inequality clarified the Dranishnikov inequality.

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“…There were partial results towards Rudyak's conjecture [8], [16] until it was settled in [5]. Later it was shown in [6] (also see the followup [13]) that the Grossmann-Whitehead type estimate holds for complexes with the fundamental group having small cohomological dimension. Namely, it was shown that cat LS X ≤ cd(π 1 (X))+dim X/2.…”

Section: Introductionmentioning

confidence: 99%