Universal Menger compacta and universal mappings

A. N. Dranishnikov

Sbornik. Mathematics, Tome 57 (1987) no. 1, pp. 131-149

Résumé

For any positive integer $n$ the author constructs a continuous mapping $f_n\colon M_n\to M_n$ of the $n$-dimensional Menger compactum onto itself that is universal in the class of mappings between $n$-dimensional compacta, i.e., for any continuous mapping $g\colon X\to Y$ between $n$-dimensional compacta there exist imbeddings of $X$ and $Y$ in $M_n$ such that the restriction of $f_n$ to $X$ is homeomorphic to $g$. The mapping $f_n$ plays the same role in the theory of Menger $n$-dimensional manifolds as the projection $\pi\colon Q\times Q\to Q$ plays in the theory of $Q$-manifolds ($Q$ is the Hilbert cube). It can be used to carry over the classical theorems in the theory of $Q$-manifolds to the theory of $M_n$-manifolds: Stabilization theorem. {\it For any $M_n$-manifold $X$ and any imbedding of $X$ in $M_n$ the space $f_n^{-1}(X)$ is homeomorphic to $X$.} Triangulation theorem. {\it For any $M_n$-manifold $X$ there exists an $n$-dimensional polyhedron $K$ such that the space $f_n^{-1}(K)$ is homeomorphic to $X$ for every imbedding of $K$ in $M_n$.} Bibliography: 20 titles.