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# 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

#### 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

Published 4 December 2014