FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

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

Ljusternik--Schnirelman theorem and β f

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 ( ≤ dimX+|G|).

As a corollary we prove that for all numbers n and k and the free action of the group G=Z2k+1 on the space G*G*...*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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