Combinatorial Methods in Topology and Algebra pp 149-153 | Cite as

# On Highly Regular Embeddings

## 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*

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 *ℓ* distinct points are skew affine subspaces. This extends work by Ghomi and Tabachnikov [8] for *ℓ* = 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## 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.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.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.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.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.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.Chisholm, M.E.:
*k*-regular mappings of 2^{n}-dimensional euclidean space. Proc. Am. Math. Soc.**74**, 187–190 (1979)Google Scholar - 7.Cohen, F.R., Handel, D.:
*k*-regular embeddings of the plane. Proc. Am. Math. Soc.**72**, 201–204 (1978)Google Scholar - 8.Ghomi, M., Tabachnikov, S.: Totally skew embeddings of manifolds. Math. Z.
**258**, 499–512 (2008)zbMATHMathSciNetCrossRefGoogle Scholar - 9.Handel, D.: Obstructions to 3-regular embeddings. Houston J. Math.
**5**, 339–343 (1979)zbMATHMathSciNetGoogle Scholar - 10.Handel, D.: Some existence and nonexistence theorems for
*k*-regular maps. Fundam. Math.**109**, 229–233 (1980)zbMATHMathSciNetGoogle Scholar - 11.Handel, D.: 2
*k*-regular maps on smooth manifolds. Proc. Am. Math. Soc.**124**, 1609–1613 (1996)zbMATHMathSciNetCrossRefGoogle Scholar - 12.Hung, N.H.V.: The mod 2 equivariant cohomology algebras of configuration spaces. Pac. J. Math.
**143**, 251–286 (1990)zbMATHCrossRefGoogle Scholar - 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.Vassiliev, V.: Spaces of functions that interpolate at any k-points. Funct. Anal. Appl.
**26**, 209–210 (1992)MathSciNetCrossRefGoogle Scholar - 15.Vassiliev, V.: On
*r*-neighbourly submanifolds in \(\mathbb{R}^{N}\). Topol. Methods Nonlinear Anal.**11**, 273–281 (1998)zbMATHMathSciNetGoogle Scholar