Borsuk-Ulam の定理

Borsuk-Ulam の定理は, 元々は, 球面から Euclid空間への \(\Z _2\)-equivariant map の存在についての定理である。 組み合せ論では結構重要な topological tool のようで, Matousek の本 [Mat03] がある。また Steinlein の解説 [Ste85; Ste93]もある。

Ziegler の AMS の Notices での解説 [Zie11] によると, Lovász が Borsuk-Ulam の定理を用いて Kneser 予想を証明した [Lov78] のが topological combinatorics の起源であると言われているようである。 もっとも, そこで Birch が [Bir59] で不動点定理を用いたのは, それよりずっと古いことも述べているが。

  • 古典的な Borsuk-Ulam の定理

例えば, ham sandwich theorem などが Borsuk-Ulam の定理から証明される。

  • ham sandwich theorem

Beyer と Zardecki による ham sandwich theorem の歴史に関する解説 [BZ04] がある。Zivaljevic の [Živ97] も見るとよい。

Musin [Mus15; MV23] によると, 離散版は Tucker’s lemma と呼ばれるもの [Tuc46; Fan52] である。

  • Tucker’s lemma

今では, Borsuk-Ulam の定理には, 各種の一般化や variation が知られている。 以下に, 目にしたものを記録した。

Parametrized version は何人かの人が考えていて, Dold [Dol88] によるものや Schick らの [Sch+11], Singh の [Sin11], Crabb と Singh の [CS18] などがある。

Gromov の waist of sphere theorem については, まず Memarian の解説 [Mem11] を見るのがよさそうである。

自由な \(\Z _2\) 作用を持つ空間で, Borsuk-Ulam 型の定理が成り立つような空間を特徴付けることも考えられている。Musin と Volovikov の[MV15] では, Bacon の [Bac66] が参照されている。

  • Borsuk-Ulam type space

関連した事柄としては, Wasserman [Was91] の定義した Borsuk-Ulam group がある。

References

[Bac66]

Philip Bacon. “Equivalent formulations of the Borsuk-Ulam theorem”. In: Canad. J. Math. 18 (1966), pp. 492–502. url: https://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.

[Bir59]

B. J. Birch. “On \(3N\) points in a plane”. In: Proc. Cambridge Philos. Soc. 55 (1959), pp. 289–293. url: https://doi.org/10.1017/s0305004100034071.

[Bou55]

D. G. Bourgin. “On some separation and mapping theorems”. In: Comment. Math. Helv. 29 (1955), pp. 199–214. url: https://doi.org/10.1007/BF02564279.

[BZ04]

W. A. Beyer and Andrew Zardecki. “The early history of the ham sandwich theorem”. In: Amer. Math. Monthly 111.1 (2004), pp. 58–61. url: http://dx.doi.org/10.2307/4145019.

[CS18]

Michael Crabb and Mahender Singh. “Some remarks on the parametrized Borsuk-Ulam theorem”. In: J. Fixed Point Theory Appl. 20.2 (2018), Paper No. 79, 14. arXiv: 1711.02397. url: https://doi.org/10.1007/s11784-018-0559-9.

[Dol83]

Albrecht Dold. “Simple proofs of some Borsuk-Ulam results”. In: Proceedings of the Northwestern Homotopy Theory Conference (Evanston, Ill., 1982). Vol. 19. Contemp. Math. Providence, RI: Amer. Math. Soc., 1983, pp. 65–69.

[Dol88]

Albrecht Dold. “Parametrized Borsuk-Ulam theorems”. In: Comment. Math. Helv. 63.2 (1988), pp. 275–285. url: http://dx.doi.org/10.1007/BF02566767.

[Dua89]

Hai Bao Duan. “Some Borsuk-Ulam-type theorems for maps from Riemannian manifolds into manifolds”. In: Proc. Roy. Soc. Edinburgh Sect. A 111.1-2 (1989), pp. 61–67. url: https://doi.org/10.1017/S0308210500025014.

[Fan52]

Ky Fan. “A generalization of Tucker’s combinatorial lemma with topological applications”. In: Ann. of Math. (2) 56 (1952), pp. 431–437.

[FH88]

Edward Fadell and Sufian Husseini. “An ideal-valued cohomological index theory with applications to Borsuk-Ulam and Bourgin-Yang theorems”. In: Ergodic Theory Dynam. Systems \(8^{*}\).Charles Conley Memorial Issue (1988), pp. 73–85. url: http://dx.doi.org/10.1017/S0143385700009342.

[Gro03]

M. Gromov. “Isoperimetry of waists and concentration of maps”. In: Geom. Funct. Anal. 13.1 (2003), pp. 178–215. url: http://dx.doi.org/10.1007/s000390300004.

[Hop44]

Heinz Hopf. “Eine Verallgemeinerung bekannter Abbildungs- und Überdeckungssätze”. In: Portugal. Math. 4 (1944), pp. 129–139.

[IM00]

Marek Izydorek and Wacław Marzantowicz. “The Borsuk-Ulam property for cyclic groups”. In: Topol. Methods Nonlinear Anal. 16.1 (2000), pp. 65–72. url: https://doi.org/10.12775/TMNA.2000.030.

[Jaw89]

Jan Jaworowski. “Maps of Stiefel manifolds and a Borsuk-Ulam theorem”. In: Proc. Edinburgh Math. Soc. (2) 32.2 (1989), pp. 271–279. url: http://dx.doi.org/10.1017/S0013091500028674.

[Lov78]

L. Lovász. “Kneser’s conjecture, chromatic number, and homotopy”. In: J. Combin. Theory Ser. A 25.3 (1978), pp. 319–324. url: http://dx.doi.org/10.1016/0097-3165(78)90022-5.

[Mar94]

Wacław Marzantowicz. “Borsuk-Ulam theorem for any compact Lie group”. In: J. London Math. Soc. (2) 49.1 (1994), pp. 195–208. url: http://dx.doi.org/10.1112/jlms/49.1.195.

[Mat03]

Jiřı́ Matoušek. Using the Borsuk-Ulam theorem. Universitext. Lectures on topological methods in combinatorics and geometry, Written in cooperation with Anders Björner and Günter M. Ziegler. Berlin: Springer-Verlag, 2003, pp. xii+196. isbn: 3-540-00362-2.

[Mem11]

Yashar Memarian. “On Gromov’s waist of the sphere theorem”. In: J. Topol. Anal. 3.1 (2011), pp. 7–36. arXiv: 0911.3972. url: https://doi.org/10.1142/S1793525311000507.

[Mus15]

Oleg R. Musin. “Extensions of Sperner and Tucker’s lemma for manifolds”. In: J. Combin. Theory Ser. A 132 (2015), pp. 172–187. arXiv: 1212.1899. url: https://doi.org/10.1016/j.jcta.2014.12.001.

[MV15]

Oleg R. Musin and Alexey Yu. Volovikov. “Borsuk-Ulam type spaces”. In: Mosc. Math. J. 15.4 (2015), pp. 749–766. arXiv: 1507.08872. url: https://doi.org/10.17323/1609-4514-2015-15-4-749-766.

[MV23]

Oleg R. Musin and Alexey Yu. Volovikov. “Borsuk–Ulam type theorems for G-spaces with applications to Tucker type lemmas”. In: J. Fixed Point Theory Appl. 25.1 (2023), Paper No. 32. arXiv: 1612. 07314. url: https://doi.org/10.1007/s11784-022-01035-7.

[PMS03]

Pedro L. Q. Pergher, Denise de Mattos, and Edivaldo L. dos Santos. “The Borsuk-Ulam theorem for general spaces”. In: Arch. Math. (Basel) 81.1 (2003), pp. 96–102. url: https://doi.org/10.1007/s00013-003-0038-3.

[Sch+11]

Thomas Schick, Robert Samuel Simon, Stanislaw Spież, and Henryk Toruńczyk. “A parametrized version of the Borsuk-Ulam theorem”. In: Bull. Lond. Math. Soc. 43.6 (2011), pp. 1035–1047. arXiv: 0709. 1774. url: https://doi.org/10.1112/blms/bdr037.

[Sin10]

Mahender Singh. “A simple proof of the Borsuk-Ulam theorem for \(\Z _p\)-actions”. In: Topology Proc. 36 (2010), pp. 249–253. arXiv: 1008. 1134.

[Sin11]

Mahender Singh. “Parametrized Borsuk-Ulam problem for projective space bundles”. In: Fund. Math. 211.2 (2011), pp. 135–147. arXiv: 0810.4669. url: https://doi.org/10.4064/fm211-2-2.

[Ste85]

H. Steinlein. “Borsuk’s antipodal theorem and its generalizations and applications: a survey”. In: Topological methods in nonlinear analysis. Vol. 95. Sém. Math. Sup. Presses Univ. Montréal, Montreal, QC, 1985, pp. 166–235.

[Ste93]

H. Steinlein. “Spheres and symmetry: Borsuk’s antipodal theorem”. In: Topol. Methods Nonlinear Anal. 1.1 (1993), pp. 15–33.

[Tuc46]

A. W. Tucker. “Some topological properties of disk and sphere”. In: Proc. First Canadian Math. Congress, Montreal, 1945. University of Toronto Press, Toronto, 1946, pp. 285–309.

[VDP11]

Daniel Vendrúscolo, Patricia E. Desideri, and Pedro L. Q. Pergher. “Some generalizations of the Borsuk-Ulam theorem”. In: Publ. Math. Debrecen 78.3-4 (2011), pp. 583–593. url: https://doi.org/10.5486/PMD.2011.4793.

[VW08]

Daniel Vendrúscolo and Peter Wong. “Jiang-type theorems for coincidences of maps into homogeneous spaces”. In: Topol. Methods Nonlinear Anal. 31.1 (2008), pp. 151–160. arXiv: math/0701702.

[Was91]

Arthur G. Wasserman. “Isovariant maps and the Borsuk-Ulam theorem”. In: Topology Appl. 38.2 (1991), pp. 155–161. url: http://dx.doi.org/10.1016/0166-8641(91)90082-W.

[Yan54]

Chung-Tao Yang. “On theorems of Borsuk-Ulam, Kakutani-Yamabe-Yujobô and Dyson. I”. In: Ann. of Math. (2) 60 (1954), pp. 262–282. url: https://doi.org/10.2307/1969632.

[Yan55]

Chung-Tao Yang. “On theorems of Borsuk-Ulam, Kakutani-Yamabe-Yujobô and Dyson. II”. In: Ann. of Math. (2) 62 (1955), pp. 271–283. url: https://doi.org/10.2307/1969681.

[Zie11]

Günter M. Ziegler. “3N colored points in a plane”. In: Notices Amer. Math. Soc. 58.4 (2011), pp. 550–557.

[Živ97]

Rade T. Živaljević. “Topological methods”. In: Handbook of discrete and computational geometry. CRC Press Ser. Discrete Math. Appl. Boca Raton, FL: CRC, 1997, pp. 209–224.