On a problem of Zambakhidze–Smirnov
Matematičeskie zametki, Tome 58 (1995) no. 1, pp. 3-11
Cet article a éte moissonné depuis la source Math-Net.Ru
We say that the action extension problem is solvable for a bicompact group $G$ if for any metric $G$-space $\mathbb X$ and for any topological embedding $c$ of the orbit space $X$ into a metric space $Y$ there exist a $G$-space $\mathbb Z$, an invariant topological embedding $b\colon X\to\mathbb Z$, and a homeomorphism $h\colon Y\to Z$ such that the diagram $$ </nomathmode><mathmode>\begin{alignedat}{2} &\mathbb X\ \xrightarrow{\hskip13mm b\hskip13mm}&&\ \mathbb Z \\ {\scriptstyle p}&\downarrow\hskip30pt&&\downarrow{\scriptstyle p} \\ &X \xrightarrow{\quad c\quad} \ Y\ \xrightarrow{\quad h\quad} &&\ Z. \end{alignedat} $$</mathmode><nomathmode> is commutative. We prove the following theorem: for a bicompact zero-dimensional group $G$, the action extension problem is solvable for the class of dense topological embeddings.
@article{MZM_1995_58_1_a0,
author = {S. M. Ageev},
title = {On~a~problem of {Zambakhidze{\textendash}Smirnov}},
journal = {Matemati\v{c}eskie zametki},
pages = {3--11},
year = {1995},
volume = {58},
number = {1},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/MZM_1995_58_1_a0/}
}
S. M. Ageev. On a problem of Zambakhidze–Smirnov. Matematičeskie zametki, Tome 58 (1995) no. 1, pp. 3-11. http://geodesic.mathdoc.fr/item/MZM_1995_58_1_a0/
[1] Bredon G., Vvedenie v teoriyu kompaktnykh grupp preobrazovanii, Nauka, M., 1980 | Zbl
[2] Ageev S. M., “O prodolzhenii deistviya”, Vestn. MGU. Ser. 1, 1992, no. 5, 20–23 | MR | Zbl
[3] Antonyan S. A., “Ob odnoi zadache L. G. Zambakhidze”, UMN, 41:5 (1986), 153–154 | MR
[4] Engelking R., Obschaya topologiya, Mir, M., 1986
[5] Pontryagin L. S., Nepreryvnye gruppy, Nauka, M., 1973