### Abstract

We study Farber's topological complexity (TC) of Davis' projective product spaces (PPS's). We show that, in many nontrivial instances, the TC of PPS's coming from at least two sphere factors is (much) lower than the dimension of the manifold. This is in marked contrast with the known situation for (usual) real projective spaces for which, in fact, the Euclidean immersion dimension and TC are two facets of the same problem. Low TC-values have been observed for infinite families of nonsimply connected spaces only for H-spaces, for finite complexes whose fundamental group has cohomological dimension at most 2, and now in this work for infinite families of PPS's. We discuss general bounds for the TC (and the Lusternik-Schnirelmann category) of PPS's, and compute these invariants for specific families of such manifolds. Some of our methods involve the use of an equivariant version of TC. We also give a characterization of the Euclidean immersion dimension of PPS's through a generalized concept of axial maps or, alternatively (in an appendix), nonsingular maps. This gives an explicit explanation of the known relationship between the generalized vector field problem and the Euclidean immersion problem for PPS's.

Original language | English |
---|---|

Pages (from-to) | 1027-1047 |

Number of pages | 21 |

Journal | Algebraic & Geometric Topology |

Volume | 13 |

Issue number | 2 |

DOIs | |

Publication status | Published - 8 Apr 2013 |

### Keywords

- Lusternik-Schnirelmann category
- lens spaces
- immersions
- robotics
- maps
- topological complexity
- projective product spaces
- Euclidean immersions of manifolds
- generalized axial maps
- equivariant motion planning

### Cite this

*Algebraic & Geometric Topology*,

*13*(2), 1027-1047. https://doi.org/10.2140/agt.2013.13.1027

**Topological complexity of motion planning in projective product spaces.** / Gonzalez, Jesus; Grant, Mark; Torres-Giese, Enrique; Xicotencatl, Miguel.

Research output: Contribution to journal › Article

*Algebraic & Geometric Topology*, vol. 13, no. 2, pp. 1027-1047. https://doi.org/10.2140/agt.2013.13.1027

}

TY - JOUR

T1 - Topological complexity of motion planning in projective product spaces

AU - Gonzalez, Jesus

AU - Grant, Mark

AU - Torres-Giese, Enrique

AU - Xicotencatl, Miguel

PY - 2013/4/8

Y1 - 2013/4/8

N2 - We study Farber's topological complexity (TC) of Davis' projective product spaces (PPS's). We show that, in many nontrivial instances, the TC of PPS's coming from at least two sphere factors is (much) lower than the dimension of the manifold. This is in marked contrast with the known situation for (usual) real projective spaces for which, in fact, the Euclidean immersion dimension and TC are two facets of the same problem. Low TC-values have been observed for infinite families of nonsimply connected spaces only for H-spaces, for finite complexes whose fundamental group has cohomological dimension at most 2, and now in this work for infinite families of PPS's. We discuss general bounds for the TC (and the Lusternik-Schnirelmann category) of PPS's, and compute these invariants for specific families of such manifolds. Some of our methods involve the use of an equivariant version of TC. We also give a characterization of the Euclidean immersion dimension of PPS's through a generalized concept of axial maps or, alternatively (in an appendix), nonsingular maps. This gives an explicit explanation of the known relationship between the generalized vector field problem and the Euclidean immersion problem for PPS's.

AB - We study Farber's topological complexity (TC) of Davis' projective product spaces (PPS's). We show that, in many nontrivial instances, the TC of PPS's coming from at least two sphere factors is (much) lower than the dimension of the manifold. This is in marked contrast with the known situation for (usual) real projective spaces for which, in fact, the Euclidean immersion dimension and TC are two facets of the same problem. Low TC-values have been observed for infinite families of nonsimply connected spaces only for H-spaces, for finite complexes whose fundamental group has cohomological dimension at most 2, and now in this work for infinite families of PPS's. We discuss general bounds for the TC (and the Lusternik-Schnirelmann category) of PPS's, and compute these invariants for specific families of such manifolds. Some of our methods involve the use of an equivariant version of TC. We also give a characterization of the Euclidean immersion dimension of PPS's through a generalized concept of axial maps or, alternatively (in an appendix), nonsingular maps. This gives an explicit explanation of the known relationship between the generalized vector field problem and the Euclidean immersion problem for PPS's.

KW - Lusternik-Schnirelmann category

KW - lens spaces

KW - immersions

KW - robotics

KW - maps

KW - topological complexity

KW - projective product spaces

KW - Euclidean immersions of manifolds

KW - generalized axial maps

KW - equivariant motion planning

U2 - 10.2140/agt.2013.13.1027

DO - 10.2140/agt.2013.13.1027

M3 - Article

VL - 13

SP - 1027

EP - 1047

JO - Algebraic & Geometric Topology

JF - Algebraic & Geometric Topology

SN - 1472-2747

IS - 2

ER -