On category, in the sense of Lusternik-Schnirelmann

James, I. M.

doi:10.1016/0040-9383(78)90002-2

Cited by 215 publications

(174 citation statements)

References 33 publications

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“…The weak category of X, wcat(X), is the least n such that ∆ n+1 * ; clearly wcat(X) ≤ cat(X). See [9] for a survey of the Lusternik-Schnirelmann category and related invariants. The strong category, or cone length, of X is the least n for which there is a sequence of cofibrations…”

Section: Introductionmentioning

confidence: 99%

The Lusternik-Schnirelmann category of 𝑆𝑝(3)

Suárez

Gómez-Tato

Strom

et al. 2003

Proc. Amer. Math. Soc.

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Section: Introductionmentioning

confidence: 99%

The Lusternik-Schnirelmann category of 𝑆𝑝(3)

Suárez

Gómez-Tato

Strom

et al. 2003

Proc. Amer. Math. Soc.

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“…They introduced what is today called the LusternikShnirelmann category of a space, denoted cat(X), as a tool to estimate the number of critical points of a smooth map. Their work was widely extended both in analysis, most notably by J. Schwartz [23] and R. Palais [21], and in topology, by R. Fox [10], T. Ganea [11], I. James [18] and many others. Today Lusternik-Schnirelmann category is a well-developed theory with many ramifications and methods of computation techniques that allow to systematically determine the category for most of the spaces that will appear in this paper -see [4].…”

Section: Prior Workmentioning

confidence: 99%

Complexity of the forward kinematic map

Pavešić

2017

Mechanism and Machine Theory

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Abstract. The main objective of this paper is to introduce a new method for qualitative analysis of various designs of robot arms. To this end we define the complexity of a map, examine its main properties and develop some methods of computation. In particular, when applied to a forward kinematic map associated to some robot arm structure, the complexity measures the inherent discontinuities that arise when one attempts to solve the motion planning problem for any set of input data. In the second part of the paper, we consider instabilities of motion planning in the proximity of singular points, and present explicit computations for several common robot arm configurations.

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“…Let us assume that TC(X) = 2n + 1. Then Corollary 7 implies cat(X) = n + 1, so by a theorem of James [17] (see also [1,Proposition 5.3]) there exists a cohomology class α ∈ H p (X; H p (X)) such that 0 = α n ∈ H np (X; H p (X)) (in fact, α is the class that corresponds to the identity under the identification…”

Section: Cohomological Estimatesmentioning

confidence: 99%

Spaces with high topological complexity

Franc

Pavešić

2014

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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Abstract. By a formula of Farber [7, Theorem 5.2] the topological complexity TC(X) of a (p − 1)-connected, m-dimensional CW-complex X is bounded above by (2m + 1)/p + 1. There are also various lower estimates for TC(X) such as the nilpotency of the ring H * (X × X, ∆(X)), and the weak and stable topological compexity wTC(X) and σTC(X) (see [10]). In general the difference between these upper and lower bounds can be arbitrarily large. In this paper we investigate spaces whose topological complexity is close to the maximal value given by Farber's formula and show that in these cases the gap between the lower and upper bounds is narrow and that TC(X) often coincides with the lower bounds.

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On category, in the sense of Lusternik-Schnirelmann

Cited by 215 publications

References 33 publications

The Lusternik-Schnirelmann category of 𝑆𝑝(3)

The Lusternik-Schnirelmann category of 𝑆𝑝(3)

Complexity of the forward kinematic map

Spaces with high topological complexity

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