The induced homology and homotopy functors on the coarse shape category

In this paper we consider some algebraic invariants of the coarse shape. We introduce functors pro*-Hn and pro*-πn relating the (pointed) coarse shape category (Sh**) Sh* to the category pro*-Grp. The category (Sh**) Sh*, which is recently constructed, is the supercategory of the (pointed) shape category (Sh*) Sh*, having all (pointed) topological spaces as objects. The category pro* -Grp is the supercategory of the category of pro-groups pro-Grp, both having the same object class. The functors pro*-Hn and pro*-πn extend standard functors pro-Hn and pro-πn which operate on (Sh*) Sh*. The full analogue of the well known Hurewicz theorem holds also in Sh**. We proved that the pro-homology (homotopy) sequence of every pair (X,A) of topological spaces, where A is normally embedded in X, is also exact in pro*-Grp. Regarding this matter the following general result is obtained: for every category C with zero-objects and kernels, the category pro-C is also a category with zero-objects and kernels, while morphisms of pro*-C generally don\u27t have kernels

Topological coarse shape groups of compact metric spaces

The shape theory and, relatively new, coarse shape theory are very useful in studying of topological spaces, as well as of the corresponding algebraic invariants, especially, shape and coarse shape groups. By using certain ultrametrics on special sets of pro- and pro*-morphisms, we topologize those groups when they refer to compact metric spaces and we get topological groups. In the shape case, they are isomorphic to recently constructed topological shape homotopy groups, while in the coarse shape case we get the coarse shape invariants, denoted by π_k*^d*(X,x0). We have proven some important properties of π_k*^d*(X,x0) and provided few interesting examples

The Hurewicz theorem in Sho* and Sho*2

The Hurewicz isomorphism theorem relating coarse shape groups and coarse shape homology groups of pointed metric continua is proved. A similar statement is proposed and proved in the category Sho*2 for relative coarse shape pro-groups and coarse shape homology pro-groups, and consequently a relative variant for relative coarse shape groups and coarse shape homology groups of pointed pairs of metric continua is given

The finite coarse shape - inverse systems approach and intrinsic approach

Given an arbitrary category (mathcal{C}), a category (pro^{*^f})-(mathcal{C}) is constructed such that the known (pro)-(mathcal{C}) category may be considered as a subcategory of (pro^{*^f})-(mathcal{C}) and that (pro^{*^f})-(mathcal{C}) may be considered as a subcategory of (pro^*)-(mathcal{C}). Analogously to the construction of the shape category (Sh_{(mathcal{C},mathcal{D})}) and the coarse category (Sh^*_{(mathcal{C},mathcal{D})}), an (abstract) finite coarse shape category (Sh^{*^f}_{(mathcal{C},mathcal{D})}) is obtained. Between these three categories appropriate faithful functors are defined. The finite coarse shape is also defined by an intrinsic approach using the notion of the (epsilon)-continuity. The isomorphism of the finite coarse shape categories obtained by these two approaches is constructed. Besides, an overview of some basic properties related to the notion of the (epsilon)-continuity is given

Computing coarse shape groups of solenoids

The coarse shape groups are new topological invariants which are (coarse) shape and homotopy invariants as well. Their structure is signicantly richer than the structure of shape groups. They provide information (especially, about compacta) even better than the homotopy pro-groups. Since nontrivial coarse shape groups, even for polyhedra, are too large, it is difficult to calculate them exactly. Herein, we give an explicit formula for computing coarse shape groups of a large class of metric compacta including solenoids

The coarse shape

Given a category C, a certain category pro*-C on inverse systems in C is constructed, such that the usual pro-category pro-C may be considered as a subcategory of pro*-C. By simulating the (abstract) shape category construction, Sh(C, D), an (abstract) coarse shape category Sh*(C, D) is obtained. An appropriate functor of the shape category to the coarse shape category exists. In the case of topological spaces, C = HTop and D = HPol or D = HANR, he corresponding realizing category for Sh* is pro*-HPol or pro*-HANR respectively. Concerning an operative characterization of a coarse shape isomorphism, a full analogue of the well known Morita lemma is proved, while in the case of inverse sequences, a useful sufficient condition is established. It is proved by examples that for C = Grp (groups) and C = HTop, the classification of inverse systems in pro*-C is strictly coarser than in pro-C. Therefore, the underlying coarse shape theory for topological spaces makes sense

On exactness of the coarse shape group sequence

The coarse shape groups are recently introduced. Given a pointed pair (X,X0,x0) and a kN, the relative coarse shape group π*k(X,X0,x0), having the standard relative shape group πk(X,X0,x0) for its subgroup, is defined. They establish a functorial relations of the topological, homotopy and (coarse) shape category to the category of groups. Therefore, the coarse shape groups are new algebraic topological, homotopy and (coarse) shape type invariants. For every pointed pair of metric compacta (X,X0,x0) and for every k>1, the boundary homomorphism ∂k*:π*k (X,X0,x0) → π*k-1 (X0,x0) = π* k-1(X0, {x0},x0) is introduced which induces a natural transformation. The corresponding sequence of the coarse shape groups is exact, although the shape sequence generally failed to be exact. This exactness makes powerful tool for computing coarse shape groups of some particular pointed pairs of metric compacta

Generalizirani Apolonijev problem

Apolonijev problem glasi: "Konstruiraj kružnicu koja dodiruje tri za- dane kružnice". U uvodnom dijelu definiramo Apolonijev problem reda n; n 2 N i na taj ga način generaliziramo. U nastavku rada posebno promatramo sve Apolonijeve probleme reda 1, 2 i 3 te odredujemo geometrijsko mjesto središta (GMS) rješenja što je posebno interesantno za Apolonijeve probleme reda 2 jer se kao GMS javljaju konike pa na takav način dobivamo alternativnu definiciju konika. Važno je istaknuti da se kao GMS rješenja Apolonijevih problema reda 3, izmedu ostaloga, javljaju presjeci 3 konike, 2 konike i pravca, koji su, protivno površnoj intuiciji, neprazni. Apolonijeve probleme reda n >=4 nije interesantno promatrati jer rješenja postoje samo u posebnim slučajevima. Takoder se iznosi ideja za daljne razmatranje stavljajući proizvoljni kut presjeka izmedu traženih kružnica i zadanih n elemenata umjesto kuta od 0 stupnjeva kojega smo u radu promatrali

The coarse shape path connectedness

The bi-pointed coarse shape category of topological spaces is constructed and the notions of a coarse shape path and coarse shape connectedness of a space are naturally introduced. It is proven that the shape path connectedness strictly implies the coarse shape path connectedness even on metrizable compacta. Furthermore, the coarse shape path connectedness on metrizable compacta reduces to ordinary connectedness