FUNDAMENTALNAYA
I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

1998, VOLUME 4, NUMBER 1, PAGES 11-38

S. A. Bogatyi

Abstract

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```
A generalization of the Aarts--Fokkink--Vermeer theorem (
```$k=1$
and the space is metrizable) is obtained. For every $k$ free
homeomorphisms of an $n$ -dimensional paracompact space onto
itself, the coloring number is not greater than $n+2k+1$ . As
an application, it is obtained that for the free action of
a finite group $G$ on a normal (finite dimensional paracompact)
space $X$ , the coloring number $LS$ and the genus $K$ of the
space are related by

$$

LS(X;G)=K(X;G)+|G|-1\ \ (\leqslant \dim X+|G|).

$$

As a corollary we prove that for all numbers $n$ and $k$ and
the free action of the group $G=\mathbb Z_{2k+1}$ on the space
$G*G*\cdots *G$ the coloring number is equal to $n+2k+1$ in the
theorem formulated above.

It is shown that for any $k$ pairwise permutable free
continuous maps of an $n$ -dimensional compact space $X$ into
itself, the coloring number does not exceed $n+2k+1$ . We
generalise one theorem proved by Steinlein (about a free
periodic homeomorphism), who gave a negative solution to
Lusternik's problem. For any free map of a compact space into
itself, the coloring number does not exceed the Hopf number
multiplied by four.

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