Hostname: page-component-848d4c4894-8kt4b Total loading time: 0 Render date: 2024-06-16T23:40:09.204Z Has data issue: false hasContentIssue false
  • English
  • Français

Coincidence Sets of Coincidence Producing Maps

Published online by Cambridge University Press:  20 November 2018

Helga Schirmer
Helga Schirmer*
Affiliation:
Carleton UniversityOttawa, Canada
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A theorem by H. Robbins shows that every closed and non-empty subset of the unit ball Bn in Euclidean n-space is the fixed point set of a self map of Bn. This result is extended to coincidence producing maps of Bn, where a map ƒ:X → Y is coincidence producing (or universal) if it has a coincidence with every map g:X → Y. The main result implies that if ƒ:Bn, Sn - 1 → Bn, Sn - 1 is coincidence producing and A⊂Bn closed and nonempty, then there exist a map ƒ': Bn, Sn - 1 → Bn, Sn - 1 and a map g: Bn → Bn such that ƒ' | Sn - 1 is homotopic to ƒ | Sn-1 and A is the coincidence set of ƒ' and g.

Keywords

Primary 55M20Secondary 54H25Coincidencesrealization of coincidence setscoincidence producing mapsuniversal mapsunit ball
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 26 , Issue 2 , 01 June 1983 , pp. 167 - 170
Copyright
Copyright © Canadian Mathematical Society 1983

References

1. Dugundji, J., Topology, Allyn and Bacon, Inc., Boston, 1966.Google Scholar
2. Holsztyński, W., Une généralisation du théorèm de Brouwer sur les points invariants, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 12 (1964), 603–606.Google Scholar
3. Holsztyński, W., Universality of the product mappings onto products of In and snakelike spaces, Fund. Math. 64 (1969), 147–155.Google Scholar
4. Holsztyński, W., On the product and composition of universal mappings of manifolds into cubes, Proc. Amer. Math. Soc. 58 (1976), 311–314.Google Scholar
5. Robbins, H., Some complements to Brouwer's fixed point theorem, Israel J. Math. 5 (1967), 225–226.Google Scholar
6. Schirmer, H., A Brouwer type coincidence theorem, Canad. Math. Bull. 9 (1966), 443–446.Google Scholar
7. Schirmer, H., Coincidence producing maps onto trees, Canad. Math. Bull. 10 (1967), 417–423.Google Scholar
8. Schirmer, H., On fixed sets of homeomorphisms of the n-ball, Israel J. Math. 7 (1969), 46–50.Google Scholar
9. Schirmer, H., Fixed point sets of continuous selfmaps, Fixed Point Theory, Proceedings, Sherbrooke, Quebec 1980, Lecture Notes in Mathematics 886, Springer-Verlag, , 1981, 417–428.Google Scholar