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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Topological obstructions to totally skew embeddings


Authors: Đorđe Baralić, Branislav Prvulović, Gordana Stojanović, Siniša Vrećica and Rade Živaljević
Journal: Trans. Amer. Math. Soc. 364 (2012), 2213-2226
MSC (2010): Primary 57R40; Secondary 55R40, 57R20
Published electronically: October 21, 2011
MathSciNet review: 2869204
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Abstract: Following Ghomi and Tabachnikov's 2008 work, we study the invariant $ N(M^n)$ defined as the smallest dimension $ N$ such that there exists a totally skew embedding of a smooth manifold $ M^n$ in $ \mathbb{R}^N$. This problem is naturally related to the question of estimating the geometric dimension of the stable normal bundle of the configuration space $ F_2(M^n)$ of ordered pairs of distinct points in $ M^n$. We demonstrate that in a number of interesting cases the lower bounds on $ N(M^n)$ obtained by this method are quite accurate and very close to the best known general upper bound $ N(M^n)\leq 4n+1$ established by Ghomi and Tabachnikov. We also provide some evidence for the conjecture that for every $ n$-dimensional, compact smooth manifold $ M^n$ $ (n>1)$,

$\displaystyle N(M^n)\leq 4n-2\alpha (n)+1.$


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Additional Information

Đorđe Baralić
Affiliation: Mathematical Institute, Serbian Academy of Sciences & Arts, Kneza Mihaila 36, p.p. 367, 11001 Belgrade, Serbia
Email: djbaralic@mi.sanu.ac.rs

Branislav Prvulović
Affiliation: Faculty of Mathematics, University of Belgrade, Studentski Trg 16, 11000 Belgrade, Serbia
Email: bane@matf.bg.ac.rs

Gordana Stojanović
Affiliation: Mathematical Institute, Serbian Academy of Sciences & Arts, Kneza Mihaila 36, p.p. 367, 11001 Belgrade, Serbia
Email: golence@gmail.com

Siniša Vrećica
Affiliation: Faculty of Mathematics, University of Belgrade, Studentski Trg 16, 11000 Belgrade, Serbia
Email: vrecica@matf.bg.ac.rs

Rade Živaljević
Affiliation: Mathematical Institute, Serbian Academy of Sciences & Arts, Kneza Mihaila 36, p.p. 367, 11001 Belgrade, Serbia
Email: rade@mi.sanu.ac.rs

DOI: http://dx.doi.org/10.1090/S0002-9947-2011-05499-1
PII: S 0002-9947(2011)05499-1
Received by editor(s): June 30, 2010
Received by editor(s) in revised form: October 18, 2010
Published electronically: October 21, 2011
Additional Notes: This research was supported by the Grants 174020 and 174034 of the Ministry for Science and Technological Development of Serbia
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.