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A Map of a Polyhedron onto a Disk

Published online by Cambridge University Press:  20 November 2018

Richard F. E. Strube
Richard F. E. Strube*
Affiliation:
Department of Mathematics, University of Western OntarioLondon, Ontario, N6A 3K7, Canada
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A map f: X → Y is said to be universal if for every map g:X → Y there exists an x ∈ X such that f(x) = g(x). In [2] W. Holsztynski observed that if B is a Boltyanskiĭ continuum (see [1]), then there exists a universal map f:B→ I2 such that the product map fxf:BxB→I2×I2 is not universal. Using this he showed that B can be replaced by a two-dimensional polyhedron. He did not, however, give a concrete example. We exhibit explicitly a two-dimensional polyhedron K and a universal map f:K→ I2 such that f×f:K×K→ I2×I2 is not universal.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 19 , Issue 4 , 01 December 1976 , pp. 483 - 485
Copyright
Copyright © Canadian Mathematical Society 1976

References

1. Boltyanskiï, V., An example of a two-dimensional compactum whose topological square is three-dimensional, Doklady Akad. Nauk SSSR (N.S.) 67 (1949), 597–599. [Amer. Math. Soc. Transl.]Google Scholar
2. Holsztyński, W., Universal mappings and fixed point theorems, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 15 (1967), 433–438.Google Scholar
3. Holsztyński, W., On the product and composition of universal mappings of manifolds into cubes, Proc. Amer. Math. Soc. 58 (1976), 311–314.Google Scholar
4. Spanier, E., Algebraic Topology, McGraw-Hill, 1966.Google Scholar