Knaster’s Problem for (Z 2) k -Symmetric Subsets of the Sphere $S^{2^{k}-1}$
Tóm tắt
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.
Tài liệu tham khảo
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)
Bárány, I., Matoušek, J.: Simultaneous partitions of measures by k-fans. Discrete Comput. Geom. 25(3), 317–334 (2001)
Borel, A.: Sur la cohomology des espaces fibrés principaux et des espace homogènes de groupes de Lie compact. Ann. Math. 57, 115–207 (1953)
Chen, W.: Counterexamples to Knaster’s conjecture. Topology 37(2), 401–405 (1998)
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)
Dyson, F.J.: Continuous functions defined on spheres. Ann. Math. 54, 534–536 (1951)
Floyd, E.E.: Real-valued mappings of spheres. Proc. Am. Math. Soc. 6, 1957–1959 (1955)
Griffiths, H.B.: The topology of square pegs in round holes. Proc. Lond. Math. Soc. 62(3), 647–672 (1991)
Grünbaum, B.: Partitions of mass-distributions and of convex bodies by hyperplanes. Pac. J. Math. 10, 1257–1261 (1960)
Guggenheimer, H.: Finite sets on curves and surfaces. Isr. J. Math. 3(2), 104–112 (1965)
Hausel, T., Makai, E. Jr., Szücs, A.: Inscribing cubes and covering by rhombic dodecahedra via equivariant topology. Mathematika 47(1–2), 371–397 (2000)
Hinrichs, A., Richter, C.: The Knaster problem: More counterexamples. Isr. J. Math. 145(1), 311–324 (2005)
Hsiang, W.Y.: Cohomology Theory of Topological Transformation Groups. Springer, Berlin (1975)
Karasev, R.N.: Inscribing a regular crosspolytope. Math. Notes (to appear) (In Russian). arXiv:0905.2671v2
Karasev, R.N.: Equipartition of a measure by (Z p )k-invariant fans. Discrete Comput. Geom. doi:10.1007/s00454-009-9138-6
Kashin, B.S., Szarek, S.J.: The Knaster problem and the geometry of high-dimensional cubes. C. R. Math. 336(11), 931–936 (2003)
Klee, V., Wagon, S.: Old and New Unsolved Problems in Plane Geometry and Number Theory. Dolciani Mathematical Expositions, The Mathematical Association of America (1996)
Knaster, B.: Problem 4. Colloq. Math. 30, 30–31 (1947)
Luke, G., Mishchenko, A.S.: Vector Bundles and Their Applications. Springer, Berlin (1998)
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)
Makeev, V.V.: Partitioning space in six parts. Vestn. Leningr. State Univ. 2, 31–34 (1988) (In Russian)
Makeev, V.V.: The Knaster problem and almost spherical sections. Math. USSR-Sb. 66(2), 431–438 (1990)
Makeev, V.V.: Inscribed and circumscribed polyhedra of a convex body. Math. Notes 55(4), 423–425 (1994)
Makeev, V.V.: Universally inscribed and outscribed polytopes. Doctor of mathematics thesis, Saint-Petersburg State University (2003)
McCleary, J.: A User’s Guide to Spectral Sequences. Cambridge University Press, Cambridge (2001)
Milnor, J., Stasheff, J.: Characteristic Classes. Princeton University Press, Princeton (1974)
Mùi, H.: Modular invariant theory and cohomology algebras of symmetric groups. J. Fac. Sci. Univ. Tokyo 22, 319–369 (1975)
Schnirelmann, L.G.: On some geometric properties of closed curves. Usp. Mat. Nauk 10, 34–44 (1944). (In Russian)
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)
Stone, A.H., Tukey, J.W.: Generalized “sandwich” theorems. Duke Math. J. 9 (1942)
Volovikov, A.Yu.: A Bourgin–Yang-type theorem for Z n p -action. Mat. Sb. 183(2), 115–144 (1992) (in Russian); translation in Russ. Acad. Sci. Sb. Math. 76(2), 361–387 (1993)
Vrećica, S.T., Živaljević, R.T.: Conical equipartitions of mass distributions. Discrete Comput. Geom. 25, 335–350 (2001)
Yamabe, H., Yujobo, Z.: On the continuous function defined on a sphere. Osaka Math. J. 2(1), 19–22 (1950)
Živaljević, R.: Topological methods. In: Goodman, J.E., O’Rourke, J. (eds.) Handbook of Discrete and Computational Geometry. CRC, Boca Raton (2004)