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Borsuk-Ulam の定理

Borsuk-Ulam , から Euclid への 2 -equivariant map についての である では topological tool のよう , Matousek [ Mat03 ] がある また Steinlein [ Ste85 Ste93 ] ある

Ziegler AMS Notices での [ Zie11 ] によると , Lovász Borsuk-Ulam いて Kneser した [ Lov78 ] のが topological combinatorics であ ると われているようである

  • 典的 Borsuk-Ulam

えば , ham sandwich theorem などが Borsuk-Ulam から れる

  • ham sandwich theorem

Beyer Zardecki による ham sandwich theorem する [ BZ04 ] があ Zivaljevic [ Živ97 ] るとよい

Musin [ Mus15 MV ] によると , Tucker’s lemma ばれるもの [ Tuc46 Fan52 ] である

  • Tucker’s lemma

では , られている

Komiya [ Kom93 ] によると , Fadell Husseini [ FH88 ] Jaworowski [ Jaw89 ] Stiefel えている

Karasev [ KHA14 ] によると , Gromov [ Gro03 ] による もある まずは Memarian [ Mem11 ] るのがよさそうである そこでは , “Waist of Sphere Theorem” ばれている

2 Borsuk-Ulam つような けることも えられている Musin Volovikov [ MV15 ] では , Bacon [ Bac66 ] されている

  • Borsuk-Ulam type space

した としては , Wasserman [ Was91 ] した Borsuk-Ulam group ある

References

[Bac66]     Philip Bacon. Equivalent formulations of the Borsuk-Ulam theorem. Canad. J. Math. , 18:492–502, 1966, http://dx.doi.org/10.4153/CJM-1966-049-9 .

[BBM]     Pavle V.M. Blagojevic, Aleksandra S. Dimitrijevic Blagojevic, and John McCleary. Borsuk-Ulam Theorems for Complements of Arrangements, arXiv:math/0612002 .

[BDaH15]     Paul F. Baum, Ludwik D  ‘    abrowski, and Piotr M. Hajac. Noncommutative Borsuk-Ulam-type conjectures. In From Poisson brackets to universal quantum symmetries , volume 106 of Banach Center Publ. , pages 9–18. Polish Acad. Sci. Inst. Math., Warsaw, 2015, arXiv:1502.05756 .

[BZ04]     W. A. Beyer and Andrew Zardecki. The early history of the ham sandwich theorem. Amer. Math. Monthly , 111(1):58–61, 2004, http://dx.doi.org/10.2307/4145019 .

[Dol83]     Albrecht Dold. Simple proofs of some Borsuk-Ulam results. In Proceedings of the Northwestern Homotopy Theory Conference (Evanston, Ill., 1982) , volume 19 of Contemp. Math. , pages 65–69. Amer. Math. Soc., Providence, RI, 1983, http://dx.doi.org/10.1090/conm/019/711043 .

[Dol88]     Albrecht Dold. Parametrized Borsuk-Ulam theorems. Comment. Math. Helv. , 63(2):275–285, 1988, http://dx.doi.org/10.1007/BF02566767 .

[Fan52]     Ky Fan. A generalization of Tucker’s combinatorial lemma with topological applications. Ann. of Math. (2) , 56:431–437, 1952.

[FH88]     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 * (Charles Conley Memorial Issue):73–85, 1988, http://dx.doi.org/10.1017/S0143385700009342 .

[Gro03]     M. Gromov. Isoperimetry of waists and concentration of maps. Geom. Funct. Anal. , 13(1):178–215, 2003, http://dx.doi.org/10.1007/s000390300002 .

[Jaw89]     Jan Jaworowski. Maps of Stiefel manifolds and a Borsuk-Ulam theorem. Proc. Edinburgh Math. Soc. (2) , 32(2):271–279, 1989, http://dx.doi.org/10.1017/S0013091500028674 .

[KHA14]     Roman Karasev, Alfredo Hubard, and Boris Aronov. Convex equipartitions: the spicy chicken theorem. Geom. Dedicata , 170:263–279, 2014, arXiv:1306.2741 .

[Kom93]     Katsuhiro Komiya. Borsuk-Ulam theorem and Stiefel manifolds. J. Math. Soc. Japan , 45(4):611–626, 1993, http://dx.doi.org/10.2969/jmsj/04540611 .

[Lov78]     L. Lovász. Kneser’s conjecture, chromatic number, and homotopy. J. Combin. Theory Ser. A , 25(3):319–324, 1978, http://dx.doi.org/10.1016/0097-3165(78)90022-5 .

[Mar94]     Wacł aw Marzantowicz. Borsuk-Ulam theorem for any compact Lie group. J. London Math. Soc. (2) , 49(1):195–208, 1994, http://dx.doi.org/10.1112/jlms/49.1.195 .

[Mat03]     Jiří Matoušek. Using the Borsuk-Ulam theorem . Universitext. Springer-Verlag, Berlin, 2003. Lectures on topological methods in combinatorics and geometry, Written in cooperation with Anders Björner and Günter M. Ziegler.

[Mem11]     Yashar Memarian. On Gromov’s waist of the sphere theorem. J. Topol. Anal. , 3(1):7–36, 2011, arXiv:0911.3972 .

[Mus15]     Oleg R. Musin. Extensions of Sperner and Tucker’s lemma for manifolds. J. Combin. Theory Ser. A , 132:172–187, 2015, arXiv:1212.1899 .

[MV]     Oleg R. Musin and Alexey Yu. Volovikov. Tucker type lemmas for G-spaces, arXiv:1612.07314 .

[MV15]     Oleg R. Musin and Alexey Yu. Volovikov. Borsuk-Ulam type spaces. Mosc. Math. J. , 15(4):749–766, 2015, arXiv:1507.08872 .

[Pas]     Benjamin Passer. A Noncommutative Borsuk-Ulam Theorem for Natsume-Olsen Spheres, arXiv:1503.01822 .

[Sin10]     Mahender Singh. A simple proof of the Borsuk-Ulam theorem for p -actions. Topology Proc. , 36:249–253, 2010, arXiv:1008.1134 .

[SSST11]     Thomas Schick, Robert Samuel Simon, Stanislaw Spież, and Henryk Toruńczyk. A parametrized version of the Borsuk-Ulam theorem. Bull. Lond. Math. Soc. , 43(6):1035–1047, 2011, arXiv:0709.1774 .

[Ste85]     H. Steinlein. Borsuk’s antipodal theorem and its generalizations and applications: a survey. In Topological methods in nonlinear analysis , volume 95 of S ém. Math. Sup. , pages 166–235. Presses Univ. Montréal, Montreal, QC, 1985.

[Ste93]     H. Steinlein. Spheres and symmetry: Borsuk’s antipodal theorem. Topol. Methods Nonlinear Anal. , 1(1):15–33, 1993.

[Tuc46]     A. W. Tucker. Some topological properties of disk and sphere. In Proc. First Canadian Math. Congress, Montreal, 1945 , pages 285–309. University of Toronto Press, Toronto, 1946.

[Was91]     Arthur G. Wasserman. Isovariant maps and the Borsuk-Ulam theorem. Topology Appl. , 38(2):155–161, 1991, http://dx.doi.org/10.1016/0166-8641(91)90082-W .

[Zie11]     Günter M. Ziegler. 3N colored points in a plane. Notices Amer. Math. Soc. , 58(4):550–557, 2011.

[Živ97]     Rade T. Živaljević. Topological methods. In Handbook of discrete and computational geometry , CRC Press Ser. Discrete Math. Appl., pages 209–224. CRC, Boca Raton, FL, 1997.