Geodesic
A Map of a Polyhedron onto a Disk
Canadian mathematical bulletin, Tome 19 (1976) no. 4, pp. 483-485

Voir la notice de l'article provenant de la source Cambridge University Press

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→ I 2 such that the product map fxf:BxB→I 2×I 2 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→ I 2 such that f×f:K×K→ I 2×I 2 is not universal. 
Strube, Richard F. E. A Map of a Polyhedron onto a Disk. Canadian mathematical bulletin, Tome 19 (1976) no. 4, pp. 483-485. doi: 10.4153/CMB-1976-072-3
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[1] 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] 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] 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] 4. Spanier, E., Algebraic Topology, McGraw-Hill, 1966. Google Scholar

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