Hopf Invariants for sectional category with applications to topological robotics

Jesús González; Mark Grant; Lucile Vandembroucq

doi:10.1093/qmath/haz019

Hopf Invariants for sectional category with applications to topological robotics

Jesús González, Mark Grant^* (Corresponding Author), Lucile Vandembroucq

^*Corresponding author for this work

Mathematical Science

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Abstract

We develop a theory of generalized Hopf invariants in the setting of sectional category. In particular we show how Hopf invariants for a product of fibrations can be identified as shuffle joins of Hopf invariants for the factors. Our results are applied to the study of Farber’s topological complexity for 2-cell complexes, as well as to the construction of a counterexample to the analogue for topological complexity of Ganea’s conjecture on Lusternik-Schnirelmann category.

Original language	English
Pages (from-to)	1209-1252
Number of pages	43
Journal	Quarterly Journal of Mathematics
Volume	70
Issue number	4
Early online date	15 Jul 2019
DOIs	https://doi.org/10.1093/qmath/haz019
Publication status	Published - Dec 2019

Keywords

Sectional category
topological complexity
generalized Hopf invariants
two-cell complexes
Ganea conjecture
join and shuffle maps

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10.1093/qmath/haz019Licence: Unspecified

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This is a pre-copyedited, author-produced version of an article accepted for publication in Quarterly Journal of Mathematics following peer review. The version of record Jesús González, Mark Grant, Lucile Vandembroucq, Hopf invariants for sectional category with applications to topological robotics, The Quarterly Journal of Mathematics, Volume 70, Issue 4, December 2019, Pages 1209–1252 is available online at: https://doi.org/10.1093/qmath/haz019
Accepted author manuscript, 414 KBLicence: Unspecified

https://arxiv.org/abs/1405.6891Licence: Unspecified

Cite this

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title = "Hopf Invariants for sectional category with applications to topological robotics",

abstract = "We develop a theory of generalized Hopf invariants in the setting of sectional category. In particular we show how Hopf invariants for a product of fibrations can be identified as shuffle joins of Hopf invariants for the factors. Our results are applied to the study of Farber{\textquoteright}s topological complexity for 2-cell complexes, as well as to the construction of a counterexample to the analogue for topological complexity of Ganea{\textquoteright}s conjecture on Lusternik-Schnirelmann category.",

keywords = "Sectional category, topological complexity, generalized Hopf invariants, two-cell complexes, Ganea conjecture, join and shuffle maps",

author = "Jes{\'u}s Gonz{\'a}lez and Mark Grant and Lucile Vandembroucq",

note = "J.C. Partially supported by Conacyt Research Grant 221221. L.V. Partially supported by Portuguese Funds through FCT (Funda¸c˜ao para a Ciˆencia e a Tecnologia) within the Project UID/MAT/00013/2013. ",

year = "2019",

month = dec,

doi = "10.1093/qmath/haz019",

language = "English",

volume = "70",

pages = "1209--1252",

journal = "Quarterly Journal of Mathematics",

issn = "0033-5606",

publisher = "Oxford University Press",

number = "4",

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T1 - Hopf Invariants for sectional category with applications to topological robotics

AU - González, Jesús

AU - Grant, Mark

AU - Vandembroucq, Lucile

N1 - J.C. Partially supported by Conacyt Research Grant 221221. L.V. Partially supported by Portuguese Funds through FCT (Funda¸c˜ao para a Ciˆencia e a Tecnologia) within the Project UID/MAT/00013/2013.

PY - 2019/12

Y1 - 2019/12

N2 - We develop a theory of generalized Hopf invariants in the setting of sectional category. In particular we show how Hopf invariants for a product of fibrations can be identified as shuffle joins of Hopf invariants for the factors. Our results are applied to the study of Farber’s topological complexity for 2-cell complexes, as well as to the construction of a counterexample to the analogue for topological complexity of Ganea’s conjecture on Lusternik-Schnirelmann category.

AB - We develop a theory of generalized Hopf invariants in the setting of sectional category. In particular we show how Hopf invariants for a product of fibrations can be identified as shuffle joins of Hopf invariants for the factors. Our results are applied to the study of Farber’s topological complexity for 2-cell complexes, as well as to the construction of a counterexample to the analogue for topological complexity of Ganea’s conjecture on Lusternik-Schnirelmann category.

KW - Sectional category

KW - topological complexity

KW - generalized Hopf invariants

KW - two-cell complexes

KW - Ganea conjecture

KW - join and shuffle maps

UR - http://www.mendeley.com/research/hopf-invariants-sectional-category-applications-topological-robotics

U2 - 10.1093/qmath/haz019

DO - 10.1093/qmath/haz019

M3 - Article

VL - 70

SP - 1209

EP - 1252

JO - Quarterly Journal of Mathematics

JF - Quarterly Journal of Mathematics

SN - 0033-5606

IS - 4

ER -

Hopf Invariants for sectional category with applications to topological robotics

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Hopf Invariants for sectional category with applications to topological robotics

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