Abstract
This is a largely expository account of various aspects of the Borsuk–Ulam theorem, including extensions of the classical theorem to families of maps parametrized by a base space and to multivalued maps. The main technical tool is the Euler class with compact supports.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Borsuk K.: Drei Sätze über die n-dimensionale Euklidische Sphäre. Fund. Math. 20, 177–190 (1933)
Conner P.E., Floyd E.E.: Differentiable Periodic Maps. Academic Press, New York (1964)
Conner P.E., Floyd E.E.: Fixed point free involutions and equivariant maps. Bull. Amer. Math. Soc. (N.S.) 66, 416–441 (1960)
Crabb M.C.: The homotopy coincidence index. J. Fixed Point Theory Appl. 7, 1–32 (2010)
Crabb M.C.: The topological Tverberg theorem and related topics. J. Fixed Point Theory Appl. 12, 1–25 (2012)
Crabb M.C., Jaworowski J.: Theorems of Kakutani and Dyson revisited. J. Fixed Point Theory Appl. 5, 227–236 (2009)
Dold A.: Parametrized Borsuk-Ulam theorems. Comment. Math. Helv. 63, 275–285 (1988)
Floyd E.E.: Real-valued mappings of spheres. Proc. Amer. Math. Soc. 6, 957–959 (1955)
Granas A., Dugundji J.: Fixed Point Theory. Springer, New York (2003)
Granas A., Jaworowski J.: Some theorems on multi-valued mappings of subsets of the Euclidean space. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astr. Phys. 7, 277–283 (1959)
Hopf H.: Eine Verallgemeinerung bekannter Abbildungs- und Überdeckungssätze. Port. Math. 4, 129–139 (1944)
Izydorek M., Jaworowski J.: Parametrized Borsuk-Ulam theorems for multivalued maps. Proc. Amer. Math. Soc. 116, 273–278 (1992)
Jaworowski J.: A continuous version of the Borsuk-Ulam theorem. Proc. Amer. Math. Soc. 82, 112–114 (1981)
Jaworowski J.: Involutions on compact spaces and a generalization of Borsuk’s theorem on antipodes. Bull. Acad. Polon. Sci. Cl. III 3, 289–292 (1955)
Jaworowski J.: Maps of Stiefel manifolds and a Borsuk-Ulam theorem. Proc. Edinb. Math. Soc. (2) 32, 271–279 (1989)
Jaworowski J.: Some consequences of the Vietoris mapping theorem. Fund. Math. 45, 261–272 (1958)
Jaworowski J.: Theorem on antipodes for multivalued mappings and a fixed point theorem. Bull. Acad. Polon. Sci. Cl. III 4, 187–192 (1956)
Jaworowski J.: The set of balanced orbits of maps of S 1 and S 3 actions. Proc. Amer. Math. Soc. 98, 158–162 (1986)
Kakutani S.: A proof that there exists a circumscribing cube around any bounded closed convex set in R 3. Ann. of Math. (2) 43, 739–741 (1942)
L. A. Lusternik and L. Schnirelmann, Topological methods in the calculus of variations. Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow, 1930 (in Russian).
Schick T., Simon R.S., Spież S., Toruńczyk H.: A parametrized version of the Borsuk-Ulam theorem. Bull. Lond. Math. Soc. 43, 1035–1047 (2011)
H. Steinlein, Borsuk’s antipodal theorem and its generalizations and applications: A survey. In: Topological Methods in Nonlinear Analysis, Presses Univ. Montréal, Montréal, QC, 1985, 166–235.
A. Yu. Volovikov, Mappings of free \({\mathbf{Z}_p}\) -spaces into manifolds. Math. USSR Izv. 20 (1983), 35–53.
Yang C.-T.: On maps from spheres to Euclidean spaces. Amer. J. Math. 79, 725–732 (1957)
Yang C.-T.: On theorems of Borsuk-Ulam, Kakutani-Yamabe-Yujobôoand Dyson, I. Ann. of Math. (2) 60, 262–282 (1954)
Author information
Authors and Affiliations
Corresponding author
Additional information
To Kazimierz Gęba
Sadly, Jan Jaworowski died before this article reached its final form. Building on Jan’s work extending over almost sixty years on various extensions of the Borsuk–Ulam theorem, this paper is presented now as a tribute to his many contributions to the theory. Jan’s first publication was a joint paper with K. Borsuk. It is a privilege for me to be the coauthor of his last paper. The responsibility for the final version, especially for any errors, is mine; had it been produced by Jan, it would certainly have been better written. MCC.
Rights and permissions
About this article
Cite this article
Crabb, M.C., Jaworowski, J. Aspects of the Borsuk–Ulam theorem. J. Fixed Point Theory Appl. 13, 459–488 (2013). https://doi.org/10.1007/s11784-013-0130-7
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11784-013-0130-7