An upper bound for topological complexity

Michael Farber; Mark Grant; Gregory Lupton; John Oprea

doi:10.1016/j.topol.2019.01.007

An upper bound for topological complexity

Michael Farber, Mark Grant, Gregory Lupton, John Oprea (Corresponding Author)

Mathematical Science

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Abstract

In [11], a new approximating invariant TCD for topological complexity was introduced called D-topological complexity. In this paper, we explore more fully the properties of TCD and the connections between TCD and invariants of Lusternik-Schnirelmann type. We also introduce a new TC-type invariant TCf that can be used to give an upper bound for TC,
TC(X) ≤ TCD (X) + 2dimX − k k + 1 ,
where X is a finite dimensional simplicial complex with k-connected universal cover X˜ . The above inequality is a refinement of an estimate given by Dranishnikov [5].

Original language	English
Pages (from-to)	109-125
Number of pages	17
Journal	Topology and its Applications
Volume	255
Early online date	30 Jan 2019
DOIs	https://doi.org/10.1016/j.topol.2019.01.007
Publication status	Published - 15 Mar 2019

Bibliographical note

This work was partially supported by a grant from the Simons Foundation: (#244393 to John Oprea).

The authors would like to thank the Mathematisches Forschungsinstitut Oberwolfach for its generosity in supporting a July 2017 Research in Pairs stay where this work was begun.

Keywords

topological complexity
Lusternik-Schnirelmann categogory
Lusternik–Schnirelmann category
Topological complexity

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Accepted Manuscript
© 2019. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/
Accepted author manuscript, 248 KBLicence: CC BY-NC-ND

https://arxiv.org/abs/1807.03994

Cite this

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title = "An upper bound for topological complexity",

abstract = "In [11], a new approximating invariant TCD for topological complexity was introduced called D-topological complexity. In this paper, we explore more fully the properties of TCD and the connections between TCD and invariants of Lusternik-Schnirelmann type. We also introduce a new TC-type invariant TCf that can be used to give an upper bound for TC,TC(X) ≤ TCD (X) + 2dimX − k k + 1 ,where X is a finite dimensional simplicial complex with k-connected universal cover X˜ . The above inequality is a refinement of an estimate given by Dranishnikov [5].",

keywords = "topological complexity, Lusternik-Schnirelmann categogory, Lusternik–Schnirelmann category, Topological complexity",

author = "Michael Farber and Mark Grant and Gregory Lupton and John Oprea",

note = "This work was partially supported by a grant from the Simons Foundation: (#244393 to John Oprea). The authors would like to thank the Mathematisches Forschungsinstitut Oberwolfach for its generosity in supporting a July 2017 Research in Pairs stay where this work was begun.",

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AU - Farber, Michael

AU - Grant, Mark

AU - Lupton, Gregory

AU - Oprea, John

N1 - This work was partially supported by a grant from the Simons Foundation: (#244393 to John Oprea). The authors would like to thank the Mathematisches Forschungsinstitut Oberwolfach for its generosity in supporting a July 2017 Research in Pairs stay where this work was begun.

PY - 2019/3/15

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N2 - In [11], a new approximating invariant TCD for topological complexity was introduced called D-topological complexity. In this paper, we explore more fully the properties of TCD and the connections between TCD and invariants of Lusternik-Schnirelmann type. We also introduce a new TC-type invariant TCf that can be used to give an upper bound for TC,TC(X) ≤ TCD (X) + 2dimX − k k + 1 ,where X is a finite dimensional simplicial complex with k-connected universal cover X˜ . The above inequality is a refinement of an estimate given by Dranishnikov [5].

AB - In [11], a new approximating invariant TCD for topological complexity was introduced called D-topological complexity. In this paper, we explore more fully the properties of TCD and the connections between TCD and invariants of Lusternik-Schnirelmann type. We also introduce a new TC-type invariant TCf that can be used to give an upper bound for TC,TC(X) ≤ TCD (X) + 2dimX − k k + 1 ,where X is a finite dimensional simplicial complex with k-connected universal cover X˜ . The above inequality is a refinement of an estimate given by Dranishnikov [5].

KW - topological complexity

KW - Lusternik-Schnirelmann categogory

KW - Lusternik–Schnirelmann category

KW - Topological complexity

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VL - 255

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JO - Topology and its Applications

JF - Topology and its Applications

ER -

An upper bound for topological complexity

Abstract

Bibliographical note

Keywords

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