Homology, Homotopy and Applications

Volume 15 (2013)

Number 2

Spaces of topological complexity one

Pages: 73 – 81

DOI: http://dx.doi.org/10.4310/HHA.2013.v15.n2.a4

Authors

Mark Grant (School of Mathematical Sciences, The University of Nottingham, United Kingdom)

Gregory Lupton (Department of Mathematics, Cleveland State University, Cleveland, Ohio, U.S.A.)

John Oprea (Department of Mathematics, Cleveland State University, Cleveland, Ohio, U.S.A.)

Abstract

We prove that a space whose topological complexity equals 1 is homotopy equivalent to some odd-dimensional sphere. We prove a similar result, although not in complete generality, for spaces $X$ whose higher topological complexity $TC_n(X)$ is as low as possible, namely $n - 1$.

Keywords

Lusternik-Schnirelmann category, topological complexity, topological robotics, acyclic space, co-H-space, homology sphere

2010 Mathematics Subject Classification

55M30, 55S40

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