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Discrete & Computational Geometry

, Volume 44, Issue 2, pp 429–438 | Cite as

Knaster’s Problem for (Z 2) k -Symmetric Subsets of the Sphere \(S^{2^{k}-1}\)

  • R. N. Karasev
Article

Abstract

We prove a Knaster-type result for orbits of the group (Z 2) k in \(S^{2^{k}-1}\) , calculating the Euler class obstruction. As a consequence, we obtain a result about inscribing skew crosspolytopes in hypersurfaces in \(\mathbb{R}^{2^{k}}\) and a result about equipartition of a measures in \(\mathbb{R}^{2^{k}}\) by (Z 2)k+1-symmetric convex fans.

Keywords

Knaster’s problem Equivariant topology Inscribing polytopes Measure partition 

References

  1. 1.
    Babenko, I.K., Bogatyi, S.A.: On the mapping of a sphere into a Euclidean space. Mat. Zametki 46(3), 3–8 (1989) (in Russian); translation in Math. Notes 46(3–4), 683–686 (1990) MathSciNetGoogle Scholar
  2. 2.
    Bárány, I., Matoušek, J.: Simultaneous partitions of measures by k-fans. Discrete Comput. Geom. 25(3), 317–334 (2001) zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Borel, A.: Sur la cohomology des espaces fibrés principaux et des espace homogènes de groupes de Lie compact. Ann. Math. 57, 115–207 (1953) CrossRefMathSciNetGoogle Scholar
  4. 4.
    Chen, W.: Counterexamples to Knaster’s conjecture. Topology 37(2), 401–405 (1998) zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Dickson, L.E.: A fundamental system of invariants of the general modular linear group with a solution of the form problem. Trans. Am. Math. Soc. 12(1), 75–98 (1911) zbMATHGoogle Scholar
  6. 6.
    Dyson, F.J.: Continuous functions defined on spheres. Ann. Math. 54, 534–536 (1951) CrossRefMathSciNetGoogle Scholar
  7. 7.
    Floyd, E.E.: Real-valued mappings of spheres. Proc. Am. Math. Soc. 6, 1957–1959 (1955) MathSciNetGoogle Scholar
  8. 8.
    Griffiths, H.B.: The topology of square pegs in round holes. Proc. Lond. Math. Soc. 62(3), 647–672 (1991) zbMATHCrossRefGoogle Scholar
  9. 9.
    Grünbaum, B.: Partitions of mass-distributions and of convex bodies by hyperplanes. Pac. J. Math. 10, 1257–1261 (1960) zbMATHGoogle Scholar
  10. 10.
    Guggenheimer, H.: Finite sets on curves and surfaces. Isr. J. Math. 3(2), 104–112 (1965) zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Hausel, T., Makai, E. Jr., Szücs, A.: Inscribing cubes and covering by rhombic dodecahedra via equivariant topology. Mathematika 47(1–2), 371–397 (2000) zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Hinrichs, A., Richter, C.: The Knaster problem: More counterexamples. Isr. J. Math. 145(1), 311–324 (2005) zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Hsiang, W.Y.: Cohomology Theory of Topological Transformation Groups. Springer, Berlin (1975) zbMATHGoogle Scholar
  14. 14.
    Karasev, R.N.: Inscribing a regular crosspolytope. Math. Notes (to appear) (In Russian). arXiv:0905.2671v2
  15. 15.
    Karasev, R.N.: Equipartition of a measure by (Z p)k-invariant fans. Discrete Comput. Geom. doi: 10.1007/s00454-009-9138-6
  16. 16.
    Kashin, B.S., Szarek, S.J.: The Knaster problem and the geometry of high-dimensional cubes. C. R. Math. 336(11), 931–936 (2003) zbMATHMathSciNetGoogle Scholar
  17. 17.
    Klee, V., Wagon, S.: Old and New Unsolved Problems in Plane Geometry and Number Theory. Dolciani Mathematical Expositions, The Mathematical Association of America (1996) Google Scholar
  18. 18.
    Knaster, B.: Problem 4. Colloq. Math. 30, 30–31 (1947) Google Scholar
  19. 19.
    Luke, G., Mishchenko, A.S.: Vector Bundles and Their Applications. Springer, Berlin (1998) zbMATHGoogle Scholar
  20. 20.
    Makeev, V.V.: Some properties of continuous mappings of spheres and problems in combinatorial geometry. In: Geometric Questions in the Theory of Functions and Sets, pp. 75–85. Kalinin Gos. Univ. (1986) Google Scholar
  21. 21.
    Makeev, V.V.: Partitioning space in six parts. Vestn. Leningr. State Univ. 2, 31–34 (1988) (In Russian) MathSciNetGoogle Scholar
  22. 22.
    Makeev, V.V.: The Knaster problem and almost spherical sections. Math. USSR-Sb. 66(2), 431–438 (1990) zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Makeev, V.V.: Inscribed and circumscribed polyhedra of a convex body. Math. Notes 55(4), 423–425 (1994) CrossRefMathSciNetGoogle Scholar
  24. 24.
    Makeev, V.V.: Universally inscribed and outscribed polytopes. Doctor of mathematics thesis, Saint-Petersburg State University (2003) Google Scholar
  25. 25.
    McCleary, J.: A User’s Guide to Spectral Sequences. Cambridge University Press, Cambridge (2001) zbMATHGoogle Scholar
  26. 26.
    Milnor, J., Stasheff, J.: Characteristic Classes. Princeton University Press, Princeton (1974) zbMATHGoogle Scholar
  27. 27.
    Mùi, H.: Modular invariant theory and cohomology algebras of symmetric groups. J. Fac. Sci. Univ. Tokyo 22, 319–369 (1975) zbMATHGoogle Scholar
  28. 28.
    Schnirelmann, L.G.: On some geometric properties of closed curves. Usp. Mat. Nauk 10, 34–44 (1944). (In Russian) Google Scholar
  29. 29.
    Steinhaus, H.: Sur la division des ensembles de l’espaces par les plans et des ensembles plans par les cercles. Fund. Math. 33, 245–263 (1945) zbMATHMathSciNetGoogle Scholar
  30. 30.
    Stone, A.H., Tukey, J.W.: Generalized “sandwich” theorems. Duke Math. J. 9 (1942) Google Scholar
  31. 31.
    Volovikov, A.Yu.: A Bourgin–Yang-type theorem for Z pn-action. Mat. Sb. 183(2), 115–144 (1992) (in Russian); translation in Russ. Acad. Sci. Sb. Math. 76(2), 361–387 (1993) zbMATHGoogle Scholar
  32. 32.
    Vrećica, S.T., Živaljević, R.T.: Conical equipartitions of mass distributions. Discrete Comput. Geom. 25, 335–350 (2001) zbMATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Yamabe, H., Yujobo, Z.: On the continuous function defined on a sphere. Osaka Math. J. 2(1), 19–22 (1950) zbMATHMathSciNetGoogle Scholar
  34. 34.
    Živaljević, R.: Topological methods. In: Goodman, J.E., O’Rourke, J. (eds.) Handbook of Discrete and Computational Geometry. CRC, Boca Raton (2004) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Dept. of MathematicsMoscow Institute of Physics and TechnologyDolgoprudnyRussia

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