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On Highly Regular Embeddings

  • Pavle V. M. Blagojević
  • Wolfgang Lück
  • Günter M. ZieglerEmail author
Part of the Springer INdAM Series book series (SINDAMS, volume 12)

Abstract

A continuous map \(\mathbb{R}^{d} \rightarrow \mathbb{R}^{N}\) 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 α(k) denotes the number of ones in the dyadic expansion of k:

For d ≥ 1 and k ≥ 1 there is no k-regular map \(\mathbb{R}^{d} \rightarrow \mathbb{R}^{N}\) for
$$\displaystyle{N < d(k -\alpha (k)) +\alpha (k).}$$

This reproduces a result of Chisholm from 1979 for the case of d being a power of 2; for the other values of d our bounds are in general better than Karasev’s [13], who had only recently gone beyond Chisholm’s special case. In particular, our lower bound turns out to be tight for k ≤ 3.

The framework of Cohen and Handel (1979) relates the existence of a k-regular map to the existence of a specific inverse of an appropriate vector bundle. Thus 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 [12] about the cohomology algebras of unordered configuration spaces.

Our study produces similar topological lower bound results also for the existence of ℓ-skew embeddings \(\mathbb{R}^{d} \rightarrow \mathbb{R}^{N}\) for which we require that the images of the tangent spaces of any distinct points are skew affine subspaces. This extends work by Ghomi and Tabachnikov [8] for  = 2.

The details for this work are provided in our paper On highly regular embeddings, Transactions of American Mathematical Society, Published electronically: May 6, 2015, http://dx.doi.org/10.1090/tran/6559.

Keywords

Vector Bundle Distinct Point Independent Vector Cohomology Algebra Moment Curve 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

The research by PVMB and GMZ has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/SFB Grant agreement no. 247029-SDModels. PVMB was also supported by the grant ON 174008 of the Serbian Ministry of Education and Science. The research by WL was funded by a DFG Leibniz Award. The work by GMZ was also supported by the DFG Collaborative Research Center TRR 109 “Discretization in Geometry and Dynamics.”

References

  1. 1.
    Blagojević, P.V.M., Ziegler, G.M.: Convex equipartitions via equivariant obstruction theory. Isr. J. Math. Journal of Topology, 8, 414–456 (2015)CrossRefGoogle Scholar
  2. 2.
    Blagojević, P.V.M., Lück, W., Ziegler, G.M.: On highly regular embeddings. Transactions of American Mathematical Society, Published electronically: May 6, 2015, http://dx.doi.org/10.1090/tran/6559
  3. 3.
    Boltjanski, V.: On imbeddings of polyhedra into Euclidean spaces. In: General Topology and its Relations to Modern Analysis and Algebra. Proceedings of a Symposium in Prague, Sept. 1961, pp. 112–114. Academia Publishing House/Czech Academy of Sciences, Prague (1962)Google Scholar
  4. 4.
    Boltjanskiĭ, V.G., Ryškov, S.S., Šaškin, J.A.: On k-regular imbeddings and their application to the theory of approximation of functions. Am. Math. Soc. Translat. 28, 211–219 (1963)Google Scholar
  5. 5.
    Borsuk, K.: On the k-independent subsets of the Euclidean space and of the Hilbert space. Bull. Acad. Polon. Sci. Cl. III. 5, 351–356 (1957)zbMATHMathSciNetGoogle Scholar
  6. 6.
    Chisholm, M.E.: k-regular mappings of 2n-dimensional euclidean space. Proc. Am. Math. Soc. 74, 187–190 (1979)Google Scholar
  7. 7.
    Cohen, F.R., Handel, D.: k-regular embeddings of the plane. Proc. Am. Math. Soc. 72, 201–204 (1978)Google Scholar
  8. 8.
    Ghomi, M., Tabachnikov, S.: Totally skew embeddings of manifolds. Math. Z. 258, 499–512 (2008)zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Handel, D.: Obstructions to 3-regular embeddings. Houston J. Math. 5, 339–343 (1979)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Handel, D.: Some existence and nonexistence theorems for k-regular maps. Fundam. Math. 109, 229–233 (1980)zbMATHMathSciNetGoogle Scholar
  11. 11.
    Handel, D.: 2k-regular maps on smooth manifolds. Proc. Am. Math. Soc. 124, 1609–1613 (1996)zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Hung, N.H.V.: The mod 2 equivariant cohomology algebras of configuration spaces. Pac. J. Math. 143, 251–286 (1990)zbMATHCrossRefGoogle Scholar
  13. 13.
    Karasev, R.N.: Regular embeddings of manifolds and topology of configuration spaces. Preprint, June 2010/2011, p. 22. http://arxiv.org/abs/1006.0613
  14. 14.
    Vassiliev, V.: Spaces of functions that interpolate at any k-points. Funct. Anal. Appl. 26, 209–210 (1992)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Vassiliev, V.: On r-neighbourly submanifolds in \(\mathbb{R}^{N}\). Topol. Methods Nonlinear Anal. 11, 273–281 (1998)zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Pavle V. M. Blagojević
    • 1
  • Wolfgang Lück
    • 2
  • Günter M. Ziegler
    • 1
    Email author
  1. 1.Institut für MathematikFreie Universität BerlinBerlinGermany
  2. 2.Mathematisches Institut der Universität BonnBonnGermany

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