# On the growth of topological complexity

On the growth of topological complexity Let TCr(X)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathrm {TC}}_r(X)$$\end{document} denote the rth topological complexity of a space X. In many cases, the generating function ∑r≥1TCr+1(X)xr\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\sum _{r\ge 1}{\mathrm {TC}}_{r+1}(X)x^r$$\end{document} is a rational function P(x)(1-x)2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\frac{P(x)}{(1-x)^2}$$\end{document} where P(x) is a polynomial with P(1)=cat(X)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$P(1)={\mathrm {cat}}(X)$$\end{document}, that is, the asymptotic growth of TCr(X)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathrm {TC}}_r(X)$$\end{document} with respect to r is cat(X)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathrm {cat}}(X)$$\end{document}. In this paper, we introduce a lower bound MTCr(X)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathrm {MTC}}_r(X)$$\end{document} of TCr(X)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathrm {TC}}_r(X)$$\end{document} for a rational space X, and estimate the growth of MTCr(X)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathrm {MTC}}_r(X)$$\end{document}. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Applied and Computational Topology Springer Journals

# On the growth of topological complexity

, Volume 4 (4) – Dec 8, 2020
8 pages

/lp/springer-journals/on-the-growth-of-topological-complexity-qpWhFjyAft
Publisher
Springer Journals
ISSN
2367-1726
eISSN
2367-1734
DOI
10.1007/s41468-020-00060-0
Publisher site
See Article on Publisher Site

### Abstract

Let TCr(X)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathrm {TC}}_r(X)$$\end{document} denote the rth topological complexity of a space X. In many cases, the generating function ∑r≥1TCr+1(X)xr\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\sum _{r\ge 1}{\mathrm {TC}}_{r+1}(X)x^r$$\end{document} is a rational function P(x)(1-x)2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\frac{P(x)}{(1-x)^2}$$\end{document} where P(x) is a polynomial with P(1)=cat(X)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$P(1)={\mathrm {cat}}(X)$$\end{document}, that is, the asymptotic growth of TCr(X)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathrm {TC}}_r(X)$$\end{document} with respect to r is cat(X)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathrm {cat}}(X)$$\end{document}. In this paper, we introduce a lower bound MTCr(X)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathrm {MTC}}_r(X)$$\end{document} of TCr(X)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathrm {TC}}_r(X)$$\end{document} for a rational space X, and estimate the growth of MTCr(X)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathrm {MTC}}_r(X)$$\end{document}.

### Journal

Journal of Applied and Computational TopologySpringer Journals

Published: Dec 8, 2020