Sequential motion planning algorithms in real projective spaces: An approach to their immersion dimension

Media type: E-Article
Title: Sequential motion planning algorithms in real projective spaces: An approach to their immersion dimension
Contributor: Cadavid-Aguilar, Natalia; González, Jesús; Gutiérrez, Darwin; Guzmán-Sáenz, Aldo; Lara, Adriana
imprint: Walter de Gruyter GmbH, 2018
Published in: Forum Mathematicum
Language: English
DOI: 10.1515/forum-2016-0231
ISSN: 1435-5337; 0933-7741
Keywords: Applied Mathematics ; General Mathematics
Origination:
Footnote:
Description: <jats:title>Abstract</jats:title> <jats:p>The <jats:italic>s</jats:italic>-th higher topological complexity <jats:inline-formula id="j_forum-2016-0231_ineq_9999_w2aab3b7b4b1b6b1aab1c17b1b3Aa"> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>TC</m:mi> <m:mi>s</m:mi> </m:msub> <m:mo>⁡</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mi>X</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_forum-2016-0231_eq_0454.png" /> <jats:tex-math>{\operatorname{TC}_{s}(X)}</jats:tex-math> </jats:alternatives> </jats:inline-formula> of a space <jats:italic>X</jats:italic> can be estimated from above by homotopical methods, and from below by homological methods. We give a thorough analysis of the gap between such estimates when <jats:inline-formula id="j_forum-2016-0231_ineq_9998_w2aab3b7b4b1b6b1aab1c17b1b7Aa"> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>X</m:mi> <m:mo>=</m:mo> <m:msup> <m:mrow> <m:mi>ℝ</m:mi> <m:mo>⁢</m:mo> <m:mi>P</m:mi> </m:mrow> <m:mi>m</m:mi> </m:msup> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_forum-2016-0231_eq_0330.png" /> <jats:tex-math>{X=\operatorname{\mathbb{R}P}^{m}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, the real projective space of dimension <jats:italic>m</jats:italic>. In particular, we describe a number <jats:inline-formula id="j_forum-2016-0231_ineq_9997_w2aab3b7b4b1b6b1aab1c17b1c11Aa"> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>r</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mi>m</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_forum-2016-0231_eq_0694.png" /> <jats:tex-math>{r(m)}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, which depends on the structure of zeros and ones in the binary expansion of <jats:italic>m</jats:italic>, and with the property that <jats:inline-formula id="j_forum-2016-0231_ineq_9996_w2aab3b7b4b1b6b1aab1c17b1c15Aa"> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mn>0</m:mn> <m:mo>≤</m:mo> <m:mrow> <m:mrow> <m:mi>s</m:mi> <m:mo>⁢</m:mo> <m:mi>m</m:mi> </m:mrow> <m:mo>-</m:mo> <m:mrow> <m:msub> <m:mi>TC</m:mi> <m:mi>s</m:mi> </m:msub> <m:mo>⁡</m:mo> <m:mrow> <m:mo>(</m:mo> <m:msup> <m:mrow> <m:mi>ℝ</m:mi> <m:mo>⁢</m:mo> <m:mi>P</m:mi> </m:mrow> <m:mi>m</m:mi> </m:msup> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:mrow> <m:mo>≤</m:mo> <m:mrow> <m:msub> <m:mi>δ</m:mi> <m:mi>s</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mi>m</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_forum-2016-0231_eq_0228.png" /> <jats:tex-math>{0\leq sm-\operatorname{TC}_{s}(\operatorname{\mathbb{R}P}^{m})\leq\delta_{s}(% m)}</jats:tex-math> </jats:alternatives> </jats:inline-formula> for <jats:inline-formula id="j_forum-2016-0231_ineq_9995_w2aab3b7b4b1b6b1aab1c17b1c17Aa"> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>s</m:mi> <m:mo>≥</m:mo> <m:mrow> <m:mi>r</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mi>m</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_forum-2016-0231_eq_0726.png" /> <jats:tex-math>{s\geq r(m)}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, where <jats:inline-formula id="j_forum-2016-0231_ineq_9994_w2aab3b7b4b1b6b1aab1c17b1c19Aa"> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:msub> <m:mi>δ</m:mi> <m:mi>s</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mi>m</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mn>0</m:mn> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_forum-2016-0231_eq_0382.png" /> <jats:tex-math>{\delta_{s}(m)=(0,1,0)}</jats:tex-math> </jats:alternatives> </jats:inline-formula> for <jats:inline-formula id="j_forum-2016-0231_ineq_9993_w2aab3b7b4b1b6b1aab1c17b1c21Aa"> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>m</m:mi> <m:mo>≡</m:mo> <m:mrow> <m:mrow> <m:mo>(</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mn>2</m:mn> <m:mo>)</m:mo> </m:mrow> <m:mo>mod</m:mo> <m:mn>4</m:mn> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_forum-2016-0231_eq_0650.png" /> <jats:tex-math>{m\equiv(0,1,2)\bmod 4}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. Such an estimation for <jats:inline-formula id="j_forum-2016-0231_ineq_9992_w2aab3b7b4b1b6b1aab1c17b1c23Aa"> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>TC</m:mi> <m:mi>s</m:mi> </m:msub> <m:mo>⁡</m:mo> <m:mrow> <m:mo>(</m:mo> <m:msup> <m:mrow> <m:mi>ℝ</m:mi> <m:mo>⁢</m:mo> <m:mi>P</m:mi> </m:mrow> <m:mi>m</m:mi> </m:msup> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_forum-2016-0231_eq_0460.png" /> <jats:tex-math>{\operatorname{TC}_{s}(\operatorname{\mathbb{R}P}^{m})}</jats:tex-math> </jats:alternatives> </jats:inline-formula> appears to be closely related to the determination of the Euclidean immersion dimension of <jats:inline-formula id="j_forum-2016-0231_ineq_9991_w2aab3b7b4b1b6b1aab1c17b1c25Aa"> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mi>ℝ</m:mi> <m:mo>⁢</m:mo> <m:mi>P</m:mi> </m:mrow> <m:mi>m</m:mi> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_forum-2016-0231_eq_0477.png" /> <jats:tex-math>{\operatorname{\mathbb{R}P}^{m}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. We illustrate the phenomenon in the case <jats:inline-formula id="j_forum-2016-0231_ineq_9990_w2aab3b7b4b1b6b1aab1c17b1c27Aa"> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>m</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mn>3</m:mn> <m:mo>⋅</m:mo> <m:msup> <m:mn>2</m:mn> <m:mi>a</m:mi> </m:msup> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_forum-2016-0231_eq_0643.png" /> <jats:tex-math>{m=3\cdot 2^{a}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. In addition, we show that, for large enough <jats:italic>s</jats:italic> and even <jats:italic>m</jats:italic>, <jats:inline-formula id="j_forum-2016-0231_ineq_9989_w2aab3b7b4b1b6b1aab1c17b1c33Aa"> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>TC</m:mi> <m:mi>s</m:mi> </m:msub> <m:mo>⁡</m:mo> <m:mrow> <m:mo>(</m:mo> <m:msup> <m:mrow> <m:mi>ℝ</m:mi> <m:mo>⁢</m:mo> <m:mi>P</m:mi> </m:mrow> <m:mi>m</m:mi> </m:msup> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_forum-2016-0231_eq_0460.png" /> <jats:tex-math>{\operatorname{TC}_{s}(\operatorname{\mathbb{R}P}^{m})}</jats:tex-math> </jats:alternatives> </jats:inline-formula> is characterized as the smallest positive integer <jats:inline-formula id="j_forum-2016-0231_ineq_9988_w2aab3b7b4b1b6b1aab1c17b1c35Aa"> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>t</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mi>t</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mi>m</m:mi> <m:mo>,</m:mo> <m:mi>s</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_forum-2016-0231_eq_0735.png" /> <jats:tex-math>{t=t(m,s)}</jats:tex-math> </jats:alternatives> </jats:inline-formula> for which there is a suitable equivariant map from Davis’ projective product space <jats:inline-formula id="j_forum-2016-0231_ineq_9987_w2aab3b7b4b1b6b1aab1c17b1c37Aa"> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>P</m:mi> <m:msub> <m:mi>𝐦</m:mi> <m:mi>s</m:mi> </m:msub> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_forum-2016-0231_eq_0421.png" /> <jats:tex-math>{\mathrm{P}_{\mathbf{m}_{s}}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> to the <jats:inline-formula id="j_forum-2016-0231_ineq_9986_w2aab3b7b4b1b6b1aab1c17b1c39Aa"> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>t</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_forum-2016-0231_eq_0212.png" /> <jats:tex-math>{(t+1)}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-st join-power <jats:inline-formula id="j_forum-2016-0231_ineq_9985_w2aab3b7b4b1b6b1aab1c17b1c41Aa"> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mo>(</m:mo> <m:msup> <m:mrow> <m:mo>(</m:mo> <m:msub> <m:mi>ℤ</m:mi> <m:mn>2</m:mn> </m:msub> <m:mo>)</m:mo> </m:mrow> <m:mrow> <m:mi>s</m:mi> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> <m:mo>)</m:mo> </m:mrow> <m:mrow> <m:mi /> <m:mo>∗</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>t</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_forum-2016-0231_eq_0187.png" /> <jats:tex-math>{((\mathbb{Z}_{2})^{s-1})^{\ast(t+1)}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. This is a (partial, but conjecturally complete) generalization of the work of Farber, Tabachnikov and Yuzvinsky relating <jats:inline-formula id="j_forum-2016-0231_ineq_9984_w2aab3b7b4b1b6b1aab1c17b1c43Aa"> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>TC</m:mi> <m:mn>2</m:mn> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_forum-2016-0231_eq_0449.png" /> <jats:tex-math>{\operatorname{TC}_{2}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> to the immersion dimension of real projective spaces.</jats:p>

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