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 $P_G$ equals the set of all primes $\mathbb{P}$, where $P_G=\{p \in \mathbb{P}: \Z_{(p)}\in$ Bockstein Basis $\sigma(G)\}$. Let n in N and let K be a connected CW-complex with $\pi_n(K)\cong G$, $\pi_k(K)\cong 0$ for $0\leq k< n$. Then for every compact metrizable space X with $X\tau K$ (i.e., with $K$ an absolute extensor for $X$), there exists a compact metrizable space Z and a surjective map $\pi: Z \to X$ such that (a) $\pi$ is cell-like, (b) $\dim Z \leq n$, and (c) $Z\tau K$.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 ⊆ 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 λ-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

Hurewicz and Dranishnikov-Smith theorems for asymptotic dimension of countable approximate groups

We establish two main results for the asymptotic dimension of countable approximate groups. The first one is a Hurewicz type formula for a global morphism of countable approximate groups $f:(\Xi, \Xi^\infty) \to (\Lambda, \Lambda^\infty)$, stating that $\mathrm{asdim} \Xi \leq \mathrm{asdim} \Lambda +\mathrm{asdim} ([\mathrm{ker} f]_c)$. This is analogous to the Dranishnikov-Smith result for groups, and is relying on another Hurewicz type formula we prove, using a 6-local morphism instead of a global one. The second result is similar to the Dranishnikov-Smith theorem stating that, for a countable group $G$, $\mathrm{asdim} G$ is equal to the supremum of asymptotic dimensions of finitely generated subgroups of $G$. Our version states that, if $(\Lambda, \Lambda^\infty)$ is a countable approximate group, then $\mathrm{asdim} \Lambda$ is equal to the supremum of asymptotic dimensions of approximate subgroups of finitely generated subgroups of $\Lambda^\infty$, with these approximate subgroups contained in $\Lambda^2$.Comment: The results in this paper were previously contained in the monograph titled "Foundations of geometric approximate group theory", by M. Cordes and the two authors listed here, but they had to be taken out of that monograph in order to shorten it. arXiv admin note: substantial text overlap with arXiv:2012.1530

Simultaneous Z/p-acyclic resolutions of expanding sequences

We prove the following Theorem: Let X be a nonempty compact metrizable space, let $l_1 \leq l_2 \leq...$ be a sequence of natural numbers, and let $X_1 \subset X_2 \subset...$ be a sequence of nonempty closed subspaces of X such that for each k in N, $dim_{Z/p} X_k \leq l_k < \infty$. Then there exists a compact metrizable space Z, having closed subspaces $Z_1 \subset Z_2 \subset...$, and a surjective cell-like map $\pi: Z \to X$, such that for each k in N, (a) $dim Z_k \leq l_k$, (b) $\pi (Z_k) = X_k$, and (c) $\pi | {Z_k}: Z_k \to X_k$ is a Z/p-acyclic map. Moreover, there is a sequence $A_1 \subset A_2 \subset...$ of closed subspaces of Z, such that for each k, $dim A_k \leq l_k$, $\pi|{A_k}: A_k\to X$ is surjective, and for k in N, $Z_k\subset A_k$ and $\pi|{A_k}: A_k\to X$ 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 članku uvodimo osnovne koncepte geometrijske teorije grupa: opisujemo kako grupu možemo shvatiti kao geometrijski objekt (Cayleyev graf) te kako na grupi uvodimo metriku. Također definiramo pojam kvaziizometrije između metričkih prostora, pa ga koristimo između grupa i njihovih Cayleyevih grafova, te između grupa i prostora. Navodimo i vrlo važan rezultat u geometrijskoj teoriji grupa – Švarc-Milnorovu lemu

Geometrija na grupama

U članku uvodimo osnovne koncepte geometrijske teorije grupa: opisujemo kako grupu možemo shvatiti kao geometrijski objekt (Cayleyev graf) te kako na grupi uvodimo metriku. Također definiramo pojam kvaziizometrije između metričkih prostora, pa ga koristimo između grupa i njihovih Cayleyevih grafova, te između grupa i prostora. Navodimo i vrlo važan rezultat u geometrijskoj teoriji grupa – Švarc-Milnorovu lemu

The significance of material traces in proving arsons on cars

У овом раду осврнућемо се на пожаре возила који су изазвани умишљајном људском делатношћу. Учиниоци ових кривичниних дела обично се oдлучују да пожар подметну преко поклопца мотора, задњих врата и крова проливањем бензина преко возила и паљењем ужареном материјом или отвореним пламеном, али у криминалистичкој праки нису ретки ни случајеви убацивања запаљиве направе у унутрашњост возила, подметање пожара испод мотора или задњег дела возила. У криминалистичким истрагама у циљу утврђивања узрока пожара најбитнији су материјални докази, док су искази лица веома ретки и непоуздани. Циљ овог рада је да укаже на специфичне материјалне трагове у случајевима намерно запаљених аутомобула, као што су: два центра пожара, плочасто пуцање ветробранског и других стакала, сливање запаљиве течности и разливање запаљиве течности, и њихов значај у поступку откривања и доказивања кривичних дела. Ови трагови несумњиво упућују на закључак да је пожар намерно изазван, те да је учинилац приликом извршења кривичног дела користио неки од поспешивача горења. Основна њихова особина је да су постојани, уочљиви и када је возило у потпуности изгорело, а на поједине од њих уопште не могу утицати ни лоше атмосверске прилике ни радње ватрогасаца приликом гашења пожара.Na nasl. str. : Srpsko udruženje za krivičnopravnu teoriju i praksu, LXI redovno godišnje savetovanje udruženja, septembar 2022