Algebraic & Geometric Topology

A mapping theorem for topological complexity

Mark Grant, Gregory Lupton, and John Oprea

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We give new lower bounds for the (higher) topological complexity of a space in terms of the Lusternik–Schnirelmann category of a certain auxiliary space. We also give new lower bounds for the rational topological complexity of a space, and more generally for the rational sectional category of a map, in terms of the rational category of a certain auxiliary space. We use our results to deduce consequences for the global (rational) homotopy structure of simply connected hyperbolic finite complexes.

Article information

Algebr. Geom. Topol., Volume 15, Number 3 (2015), 1643-1666.

Received: 29 April 2014
Revised: 21 October 2014
Accepted: 23 October 2014
First available in Project Euclid: 16 November 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55M30: Ljusternik-Schnirelman (Lyusternik-Shnirelʹman) category of a space 55P62: Rational homotopy theory
Secondary: 55S40: Sectioning fiber spaces and bundles 55Q15: Whitehead products and generalizations

Lusternik–Schnirelmann category sectional category topological complexity topological robotics sectioned fibration connective cover Avramov–Félix conjecture


Grant, Mark; Lupton, Gregory; Oprea, John. A mapping theorem for topological complexity. Algebr. Geom. Topol. 15 (2015), no. 3, 1643--1666. doi:10.2140/agt.2015.15.1643.

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