Parametrized Borsuk-Ulam theorems for multivalued maps

Izydorek, Marek; Jaworowski, Jan

doi:10.1090/S0002-9939-1992-1112493-0

Parametrized Borsuk-Ulam theorems for multivalued maps
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by Marek Izydorek and Jan Jaworowski PDF

Proc. Amer. Math. Soc. 116 (1992), 273-278 Request permission

Abstract:

By combining parametrized Borsuk-Ulam theorems proved by Dold with methods using the Vietoris mapping theorem we show that Dold’s results can be extended to multivalued maps. Such methods were invented by Eilenberg and Montgomery who applied them to multivalued fixed-point theorems, and they were used by Jaworowski to prove a multivalued version of the Borsuk-Ulam theorem. Subsequently they were extended and refined in various ways by Górniewicz and others. We also indicate how our results can be proved in the relative case, for pairs of spaces rather than for single spaces only. This allows us to obtain positive results for bundles over manifolds with boundary; for instance, over a closed interval.

References

Albrecht Dold, Parametrized Borsuk-Ulam theorems, Comment. Math. Helv. 63 (1988), no. 2, 275–285. MR 948782, DOI 10.1007/BF02566767
Samuel Eilenberg and Deane Montgomery, Fixed point theorems for multi-valued transformations, Amer. J. Math. 68 (1946), 214–222. MR 16676, DOI 10.2307/2371832

Cohomological index theory with applications to critical point theory and Borsuk-Ulam theorems

Homological methods in fixed point theory for multivalued maps

129

Marek Izydorek, Remarks on Borsuk-Ulam theorem for multivalued maps, Bull. Polish Acad. Sci. Math. 35 (1987), no. 7-8, 501–504 (English, with Russian summary). MR 939013
J. W. Jaworowski, Theorem on antipodes for multivalued mappings and a fixed point theorem, Bull. Acad. Polon. Sci. Cl. III. 4 (1956), 187–192. MR 0079270
Jan Jaworowski, A continuous version of the Borsuk-Ulam theorem, Proc. Amer. Math. Soc. 82 (1981), no. 1, 112–114. MR 603612, DOI 10.1090/S0002-9939-1981-0603612-3
Jan Jaworowski, Fibre preserving maps of sphere bundles into vector space bundles, Fixed point theory (Sherbrooke, Que., 1980) Lecture Notes in Math., vol. 886, Springer, Berlin-New York, 1981, pp. 154–162. MR 643004
Minoru Nakaoka, Equivariant point theorems for fibre-preserving maps, Osaka J. Math. 21 (1984), no. 4, 809–815. MR 765357