Parametrized Borsuk-Ulam theorems for multivalued maps

Authors:
Marek Izydorek and Jan Jaworowski

Journal:
Proc. Amer. Math. Soc. **116** (1992), 273-278

MSC:
Primary 55M20; Secondary 54C60, 55R25

DOI:
https://doi.org/10.1090/S0002-9939-1992-1112493-0

MathSciNet review:
1112493

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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.

**[D]**Albrecht Dold,*Parametrized Borsuk-Ulam theorems*, Comment. Math. Helv.**63**(1988), no. 2, 275–285. MR**948782**, https://doi.org/10.1007/BF02566767**[EM]**Samuel Eilenberg and Deane Montgomery,*Fixed point theorems for multi-valued transformations*, Amer. J. Math.**68**(1946), 214–222. MR**0016676**, https://doi.org/10.2307/2371832**[FH]**E. R. Fadell and S. Y. Husseini,*Cohomological index theory with applications to critical point theory and Borsuk-Ulam theorems*, Sém. Math. Sup., vol. 108, Université de Montréal, Montréal, 1989, pp. 10-54.**[G]**L. Górniewicz,*Homological methods in fixed point theory for multivalued maps*, Dissertationes Math.**129**(1976), 1-71.**[I]**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****[Jl]**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****[J2]**Jan Jaworowski,*A continuous version of the Borsuk-Ulam theorem*, Proc. Amer. Math. Soc.**82**(1981), no. 1, 112–114. MR**603612**, https://doi.org/10.1090/S0002-9939-1981-0603612-3**[J3]**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****[N]**Minoru Nakaoka,*Equivariant point theorems for fibre-preserving maps*, Osaka J. Math.**21**(1984), no. 4, 809–815. MR**765357**

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DOI:
https://doi.org/10.1090/S0002-9939-1992-1112493-0

Article copyright:
© Copyright 1992
American Mathematical Society