An upper bound for topological complexity

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 TC...

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Bibliographic Details
Published inTopology and its applications Vol. 255; pp. 109 - 125
Main Authors Farber, Michael, Grant, Mark, Lupton, Gregory, Oprea, John
Format Journal Article
LanguageEnglish
Published Elsevier B.V 15.03.2019
Subjects
Lusternik–Schnirelmann category
Topological complexity
secondary
Lusternik–Schnirelmann category
Topological complexity
primary
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Summary: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 TC˜ that can be used to give an upper bound for TC,(1)TC(X)≤TCD(X)+TC˜(X). This then entails a connectivity-dimension estimateTC(X)≤TCD(X)+⌈2dimX−kk+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].
ISSN:0166-8641
1879-3207
DOI:10.1016/j.topol.2019.01.007