The degree of multivalued maps from manifolds to spheres. (English) Zbl 1260.55006
Summary: In this paper, for closed connected oriented manifolds \(M\) and \(N\) of the same dimension, we study the degree of a triple ({\(\Gamma\)}, \(p\), \(q\)), where \(p\) is a Vietoris map from a compact space {\(\Gamma\)} to \(M\) and \(q\) is a continuous map from {\(\Gamma\)} to \(N\). In particular, we have Borsuk-Ulam-type degree theorems on manifolds with involutions.
MSC:
| 55M25 | Degree, winding number |
| 47H10 | Fixed-point theorems |
| 54C60 | Set-valued maps in general topology |
References:
| [1] | T. tom Dieck, Transformation Groups. de Gruyter Studies in Mathematics 8, Walter de Gruyter & Co., Berlin, 1987. · Zbl 0611.57002 |
| [2] | L. Górniewicz, Topological Fixed Point Theory of Multivalued Mappings. Mathematics and Its Applications 495, Kluwer, Dordrecht, 1999. · Zbl 0937.55001 |
| [3] | Górniewicz L., Granas A.: Fixed point theorems for multi-valued mappings of the absolute neighbourhood retracts. J. Math. Pures Appl. (9) 49, 381–395 (1970) · Zbl 0203.25203 |
| [4] | Górniewicz L., Granas A.: Some general theorems in coincidence theory. I. J. Math. Pures Appl. (9) 60, 361–373 (1981) · Zbl 0482.55002 |
| [5] | Granas A., Jaworowski J.W.: Some theorems on multi-valued mappings of subsets of the Euclidean space. Bull. Acad. Polon. Sci. 7, 277–283 (1959) · Zbl 0089.17902 |
| [6] | A. Granas and J. Dugundji, Fixed Point Theory. Springer Monographs in Mathematics, Springer-Verlag, New York, 2003. · Zbl 1025.47002 |
| [7] | Jaworowski J.W.: Theorem on antipodes for multivalued mappings and a fixed point theorem. Bull. Acad. Polon. Sci. Cl. III. 4, 187–192 (1956) · Zbl 0074.38305 |
| [8] | Jaworowski J.W.: Some consequences of the Vietoris mapping theorem. Fund. Math. 45, 261–272 (1958) · Zbl 0080.38102 |
| [9] | W. Kryszewski, Topological and approximation methods of degree theory of setvalued maps. Dissertationes Math. (Rozprawy Mat.) 336 (1994), 101 pp. · Zbl 0832.55004 |
| [10] | Spanier E.: Algebraic Topology. Springer-Verlag, New York (1966) · Zbl 0145.43303 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
