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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
Published electronically: May 6, 2015
MathSciNet review: 3449261
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Abstract | References | Similar Articles | Additional Information

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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  • [1] Alejandro Adem and R. James Milgram, Cohomology of finite groups, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 309, Springer-Verlag, Berlin, 2004. MR 2035696 (2004k:20109)
  • [2] ore Baralić, Branislav Prvulović, Gordana Stojanović, Siniša Vrećica, and Rade Živaljević, Topological obstructions to totally skew embeddings, Trans. Amer. Math. Soc. 364 (2012), no. 4, 2213-2226. MR 2869204,
  • [3] P. V. M. Blagojević, W. Lück, G. M. Ziegler, Equivariant topology of configuration spaces, J. Topology, appeared online March 21, 2015, doi: 10.1112/jtopol/jtv002.
  • [4] Pavle V. M. Blagojević and Günter M. Ziegler, The ideal-valued index for a dihedral group action, and mass partition by two hyperplanes, Topology Appl. 158 (2011), no. 12, 1326-1351. MR 2812486 (2012i:52051),
  • [5] Pavle V. M. Blagojević and Günter M. Ziegler, Convex equipartitions via equivariant obstruction theory, Israel J. Math. 200 (2014), no. 1, 49-77. MR 3219570,
  • [6] V. G. Boltjanskiĭ, S. S. Ryškov, and Ju. A. Šaškin, On $ k$-regular imbeddings and their application to the theory of approximation of functions, Amer. Math. Soc. Transl. (2) 28 (1963), 211-219. MR 0154031 (27 #3991)
  • [7] K. Borsuk, On the $ k$-independent subsets of the Euclidean space and of the Hilbert space, Bull. Acad. Polon. Sci. Cl. III. 5 (1957), 351-356, XXIX (English, with Russian summary). MR 0088710 (19,567d)
  • [8] Michael E. Chisholm, $ k$-regular mappings of $ 2^{n}$-dimensional Euclidean space, Proc. Amer. Math. Soc. 74 (1979), no. 1, 187-190. MR 521896 (82h:55022),
  • [9] F. R. Cohen, The homology of $ C_{n+1}$-spaces, $ n\geq 0$, in ``The Homology of Iterated Loop Spaces'', Lecture Notes in Math. 533, Springer, Heidelberg, 1976, pp. 207-351. MR 0436146 (55 #9096)
  • [10] F. R. Cohen and D. Handel, $ k$-regular embeddings of the plane, Proc. Amer. Math. Soc. 72 (1978), no. 1, 201-204. MR 524347 (80e:57033),
  • [11] F. R. Cohen, M. E. Mahowald, and R. J. Milgram, The stable decomposition for the double loop space of a sphere, Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976) Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, R.I., 1978, pp. 225-228. MR 520543 (80j:55009)
  • [12] F. R. Cohen and L. R. Taylor, On the representation theory associated to the cohomology of configuration spaces, Algebraic topology (Oaxtepec, 1991) Contemp. Math., vol. 146, Amer. Math. Soc., Providence, RI, 1993, pp. 91-109. MR 1224909 (94i:57057),
  • [13] Edward Fadell and Sufian Husseini, An ideal-valued cohomological index theory with applications to Borsuk-Ulam and Bourgin-Yang theorems, Ergodic Theory Dynam. Systems 8$ ^*$ (1988), no. Charles Conley Memorial Issue, 73-85. MR 967630 (89k:55002),
  • [14] Mohammad Ghomi and Serge Tabachnikov, Totally skew embeddings of manifolds, Math. Z. 258 (2008), no. 3, 499-512. MR 2369041 (2008m:57068),
  • [15] David Handel, Obstructions to $ 3$-regular embeddings, Houston J. Math. 5 (1979), no. 3, 339-343. MR 559974 (83c:57008)
  • [16] David Handel, Some existence and nonexistence theorems for $ k$-regular maps, Fund. Math. 109 (1980), no. 3, 229-233. MR 597069 (82f:57018)
  • [17] David Handel, $ 2k$-regular maps on smooth manifolds, Proc. Amer. Math. Soc. 124 (1996), no. 5, 1609-1613. MR 1307524 (96g:57025),
  • [18] David Handel and Jack Segal, On $ k$-regular embeddings of spaces in Euclidean space, Fund. Math. 106 (1980), no. 3, 231-237. MR 584495 (81h:57005)
  • [19] Dale Husemoller, Fibre bundles, 3rd ed., Graduate Texts in Mathematics, vol. 20, Springer-Verlag, New York, 1994. MR 1249482 (94k:55001)
  • [20] Nguyên H. V. Hung, The mod $ 2$ equivariant cohomology algebras of configuration spaces, Pacific J. Math. 143 (1990), no. 2, 251-286. MR 1051076 (91k:55017)
  • [21] R. N. Karasev, Regular embeddings of manifolds and topology of configuration spaces, Preprint, ver. 3, June 2011, 22 pages;
  • [22] Stanley O. Kochman, Homology of the classical groups over the Dyer-Lashof algebra, Trans. Amer. Math. Soc. 185 (1973), 83-136. MR 0331386 (48 #9719)
  • [23] Edouard Lucas, Theorie des Fonctions Numeriques Simplement Periodiques, Amer. J. Math. 1 (1878), no. 2, 184-196 (French). MR 1505161,
  • [24] Ib Madsen and R. James Milgram, The classifying spaces for surgery and cobordism of manifolds, Annals of Mathematics Studies, vol. 92, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1979. MR 548575 (81b:57014)
  • [25] John W. Milnor and James D. Stasheff, Characteristic classes, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1974. Annals of Mathematics Studies, No. 76. MR 0440554 (55 #13428)
  • [26] Howard Osborn, Vector bundles. Vol. 1: Foundations and Stiefel-Whitney classes, Pure and Applied Mathematics, vol. 101, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1982. MR 698075 (85e:55001)
  • [27] Fridolin Roth, On the category of Euclidean configuration spaces and associated fibrations, Groups, homotopy and configuration spaces, Geom. Topol. Monogr., vol. 13, Geom. Topol. Publ., Coventry, 2008, pp. 447-461. MR 2508218 (2010m:55003),
  • [28] Gordana Stojanović, Embeddings with multiple regularity, Geom. Dedicata 123 (2006), 1-10. MR 2299723 (2008c:53005),
  • [29] V. A. Vassiliev, Spaces of functions that interpolate at any $ k$-points, Funktsional. Anal. i Prilozhen. 26 (1992), no. 3, 72-74 (Russian); English transl., Funct. Anal. Appl. 26 (1992), no. 3, 209-210. MR 1189026 (94a:58024),
  • [30] Victor A. Vassiliev, On $ r$-neighbourly submanifolds in $ {\bf R}^N$, Topol. Methods Nonlinear Anal. 11 (1998), no. 2, 273-281. MR 1659458 (2000f:57020)

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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

Wolfgang Lück
Affiliation: Mathematisches Institut der Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany

Günter M. Ziegler
Affiliation: Institut für Mathematik, FU Berlin, Arnimallee 2, 14195 Berlin, Germany

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”.
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