Topological obstructions to totally skew embeddings

Baralić, Đorđe; Prvulović, Branislav; Stojanović, Gordana; Vrećica, Siniša; Živaljević, Rade

doi:10.1090/S0002-9947-2011-05499-1

Topological obstructions to totally skew embeddings
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by Đorđe Baralić, Branislav Prvulović, Gordana Stojanović, Siniša Vrećica and Rade Živaljević

Trans. Amer. Math. Soc. 364 (2012), 2213-2226

DOI: https://doi.org/10.1090/S0002-9947-2011-05499-1

Published electronically: October 21, 2011

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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 $M^n$ $(n>1)$, \[ N(M^n)\leq 4n-2\alpha (n)+1.\]

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