12 research outputs found
Bockstein basis and resolution theorems in extension theory
We prove a generalization of the Edwards-Walsh Resolution Theorem:
Theorem: Let G be an abelian group for which equals the set of all
primes , where Bockstein
Basis . Let n in N and let K be a connected CW-complex with
, for . Then for every compact
metrizable space X with (i.e., with an absolute extensor for
), there exists a compact metrizable space Z and a surjective map such that (a) is cell-like, (b) , and (c) .Comment: 23 page
The (largest) Lebesgue number and its relative version
In this paper we compare different definitions of the (largest) Lebesgue number of a cover U for a metric space X. We also introduce the relative version for the Lebesgue number of a covering family U for a subset A ā X, and justify the relevance of introducing it by giving a corrected statement and proof of the Lemma 3.4 from the paper by Buyalo and Lebedeva (2007), involving Ī»-quasi homothetic maps with coefficient R between metric spaces and the comparison of the mesh and the Lebesgue number of a covering family for a subset on both sides of the map
A proof of The Edwards-Walsh Resolution Theorem without Edwards-Walsh CW-complexes
In the paper titled "Bockstein basis and resolution theorems in extension
theory" (arXiv:0907.0491v2), we stated a theorem that we claimed to be a
generalization of the Edwards-Walsh resolution theorem. The goal of this note
is to show that the main theorem from (arXiv:0907.0491v2) is in fact equivalent
to the Edwards-Walsh resolution theorem, and also that it can be proven without
using Edwards-Walsh complexes. We conclude that the Edwards-Walsh resolution
theorem can be proven without using Edwards-Walsh complexes.Comment: 5 page
Simultaneous Z/p-acyclic resolutions of expanding sequences
We prove the following
Theorem: Let X be a nonempty compact metrizable space, let be a sequence of natural numbers, and let
be a sequence of nonempty closed subspaces of X such that for each k in N,
. Then there exists a compact metrizable space
Z, having closed subspaces , and a surjective
cell-like map , such that for each k in N,
(a) ,
(b) , and
(c) is a Z/p-acyclic map.
Moreover, there is a sequence of closed
subspaces of Z, such that for each k, ,
is surjective, and for k in N, and is a
UV^{l_k-1}-map.
It is not required that X be the union of all X_k, nor that Z be the union of
all Z_k. This result generalizes the Z/p-resolution theorem of A. Dranishnikov,
and runs parallel to a similar theorem of S. Ageev, R. Jim\'enez, and L. Rubin,
who studied the situation where the group was Z.Comment: 18 pages, title change in version 3, old title: "Z/p-acyclic
resolutions in the strongly countable Z/p-dimensional case
Geometrija na grupama
U Älanku uvodimo osnovne koncepte geometrijske teorije grupa: opisujemo kako grupu možemo shvatiti kao geometrijski objekt (Cayleyev graf) te kako na grupi uvodimo metriku. TakoÄer definiramo pojam kvaziizometrije izmeÄu metriÄkih prostora, pa ga koristimo izmeÄu grupa i njihovih Cayleyevih grafova, te izmeÄu grupa i prostora. Navodimo i vrlo važan rezultat u geometrijskoj teoriji grupa ā Å varc-Milnorovu lemu
Geometrija na grupama
U Älanku uvodimo osnovne koncepte geometrijske teorije grupa: opisujemo kako grupu možemo shvatiti kao geometrijski objekt (Cayleyev graf) te kako na grupi uvodimo metriku. TakoÄer definiramo pojam kvaziizometrije izmeÄu metriÄkih prostora, pa ga koristimo izmeÄu grupa i njihovih Cayleyevih grafova, te izmeÄu grupa i prostora. Navodimo i vrlo važan rezultat u geometrijskoj teoriji grupa ā Å varc-Milnorovu lemu