scholarly journals The Willmore Conjecture in the Real Projective Space

1999 ◽  
Vol 6 (5) ◽  
pp. 487-493 ◽  
Author(s):  
Antonio Ros
Keyword(s):  
Projective Space ◽  
Real Projective Space ◽  
The Real ◽  
Willmore Conjecture

scholarly journals An irreducible Heegaard diagram of the real projective 3-spacep3

2000 ◽  
Vol 23 (2) ◽  
pp. 123-129 ◽  
Author(s):  
Young Ho Im ◽  
Soo Hwan Kim
Keyword(s):  
Projective Space ◽  
Real Projective Space ◽  
Heegaard Diagrams ◽  
The Real ◽  
Heegaard Diagram ◽  
Genus 2

We give a genus 3 Heegaard diagramHof the real projective spacep3, which has no waves and pairs of complementary handles. So Negami's result that every genus 2 Heegaard diagram ofp3is reducible cannot be extended to Heegaard diagrams ofp3with genus 3.

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

2018 ◽  
Vol 30 (2) ◽  
pp. 397-417 ◽  
Author(s):  
Natalia Cadavid-Aguilar ◽  
Jesús González ◽  
Darwin Gutiérrez ◽  
Aldo Guzmán-Sáenz ◽  
Adriana Lara
Keyword(s):  
Positive Integer ◽  
Projective Space ◽  
Motion Planning ◽  
Product Space ◽  
Projective Spaces ◽  
Topological Complexity ◽  
Real Projective Space ◽  
Binary Expansion ◽  
The Real

AbstractThes-th higher topological complexity{\operatorname{TC}_{s}(X)}of a spaceXcan 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{X=\operatorname{\mathbb{R}P}^{m}}, the real projective space of dimensionm. In particular, we describe a number{r(m)}, which depends on the structure of zeros and ones in the binary expansion ofm, and with the property that{0\leq sm-\operatorname{TC}_{s}(\operatorname{\mathbb{R}P}^{m})\leq\delta_{s}(% m)}for{s\geq r(m)}, where{\delta_{s}(m)=(0,1,0)}for{m\equiv(0,1,2)\bmod 4}. Such an estimation for{\operatorname{TC}_{s}(\operatorname{\mathbb{R}P}^{m})}appears to be closely related to the determination of the Euclidean immersion dimension of{\operatorname{\mathbb{R}P}^{m}}. We illustrate the phenomenon in the case{m=3\cdot 2^{a}}. In addition, we show that, for large enoughsand evenm,{\operatorname{TC}_{s}(\operatorname{\mathbb{R}P}^{m})}is characterized as the smallest positive integer{t=t(m,s)}for which there is a suitable equivariant map from Davis’ projective product space{\mathrm{P}_{\mathbf{m}_{s}}}to the{(t+1)}-st join-power{((\mathbb{Z}_{2})^{s-1})^{\ast(t+1)}}. This is a (partial, but conjecturally complete) generalization of the work of Farber, Tabachnikov and Yuzvinsky relating{\operatorname{TC}_{2}}to the immersion dimension of real projective spaces.

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