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On highly regular embeddings


Authors: Pavle V. M. Blagojević, Wolfgang Lück and Günter M. Ziegler
Journal: Trans. Amer. Math. Soc. 368 (2016), 2891-2912
MSC (2010): Primary 55R80, 57N35, 57R20
DOI: https://doi.org/10.1090/tran/6559
Published electronically: May 6, 2015
MathSciNet review: 3449261
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Abstract: A continuous map $ \mathbb{R}^d\to \mathbb{R}^N$ is $ k$-regular if it maps any $ k$ pairwise distinct points to $ k$ linearly independent vectors. Our main result on $ k$-regular maps is the following lower bound for the existence of such maps between Euclidean spaces, in which $ \alpha (k)$ denotes the number of ones in the dyadic expansion of $ k$:

For $ d\geq 1$ and $ k\geq 1$ there is no $ k$-regular map $ \mathbb{R}^d\to \mathbb{R}^N$ for $ N<d(k-\alpha (k))+\alpha (k)$.
This reproduces a result of Cohen & Handel from 1978 for $ d=2$ and the extension by Chisholm from 1979 to the case when $ d$ is a power of $ 2$; for the other values of $ d$ our bounds are in general better than Karasev's (2010), who had only recently gone beyond Chisholm's special case. In particular, our lower bound turns out to be tight for $ k\le 3$.

A framework of Cohen & Handel (1979) relates the existence of a $ k$-regular map to the existence of a low-dimensional inverse of a certain vector bundle. Thus the non-existence of regular maps into $ \mathbb{R}^N$ for small $ N$ follows from the non-vanishing of specific dual Stiefel-Whitney classes. This we prove using the general Borsuk-Ulam-Bourgin-Yang theorem combined with a key observation by Hung (1990) about the cohomology algebras of configuration spaces.

Our study produces similar lower bound results also for the existence of $ \ell $-skew embeddings $ \mathbb{R}^d\to \mathbb{R}^N$, for which we require that the images of the tangent spaces of any $ \ell $ distinct points are skew affine subspaces. This extends work by Ghomi & Tabachnikov (2008) for the case $ \ell =2$.


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Additional Information

Pavle V. M. Blagojević
Affiliation: Mathematički Institut SANU, Knez Mihailova 36, 11001 Beograd, Serbia – and Institut für Mathematik, FU Berlin, Arnimallee 2, 14195 Berlin, Germany
Email: pavleb@mi.sanu.ac.rs; blagojevic@math.fu-berlin.de

Wolfgang Lück
Affiliation: Mathematisches Institut der Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany
Email: wolfgang.lueck@him.uni-bonn.de

Günter M. Ziegler
Affiliation: Institut für Mathematik, FU Berlin, Arnimallee 2, 14195 Berlin, Germany
Email: ziegler@math.fu-berlin.de

DOI: https://doi.org/10.1090/tran/6559
Received by editor(s): December 30, 2013
Received by editor(s) in revised form: July 20, 2014, and September 3, 2014
Published electronically: May 6, 2015
Additional Notes: The research by the first author leading to these results received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC Grant agreement no. 247029-SDModels. He was also supported by the grant ON 174008 of the Serbian Ministry of Education and Science.
The research by the second author leading to these results received funding from the Leibniz Award granted by the DFG
The research by the third author leading to these results received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC Grant agreement no. 247029-SDModels and by the DFG Collaborative Research Center TRR 109 “Discretization in Geometry and Dynamics”.
Article copyright: © Copyright 2015 American Mathematical Society

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