Topological coarse shape groups

Teorija (topološkog) oblika (vidi [23]) je dobro poznata grana topologije koja pruža općenitiji pogled na topološke prostore (kraće prostore) nego homotopska teorija, a svoje poopćenje ima u teoriji (topološkog) gruboga oblika (vidi [17]). Važne invarijante tih teorija, koje se pridružuju (punktiranim) prostorima su grupe oblika i gruboga oblika, redom. Nedavno je grupa oblika topologizirana u [26], te je dobivena topološka grupa, a na sličan način ćemo obogatiti i strukturu grupe gruboga oblika. U prvom dijelu rada proučavamo različite (netrivijalne) topologije na skupovima morfizama u kategorijama \(inv\mathcal{-C}, inv^\ast\mathcal{-C}\), gdje je \(\mathcal{C}\) proizvoljna kategorija. Potom, pomoću prethodno spomenutih topologija, na prirodan način topologiziramo skupove morfizama u kategorijama \(pro\mathcal{-C}, pro^\ast\mathcal{-C}\) između dvaju fiksnih objekata (tj. inverznih sustava u \(\mathcal{C}\)), preciznije, u oba slučaja uvodimo po dvije topologije, koje u \(pro-\) slučaju označavamo sa \(\mathcal{T}_{ind}\) i \(\mathcal{T}_{card}\), a u \(pro\ast-\) slučaju sa \(\mathcal{T}^\ast_{ind}\) i \(\mathcal{T}^\ast_{card}\). Pokazujemo da sve četiri spomenute topologije imaju baze čiji su elementi otvoreno-zatvoreni skupovi, te da ih smijemo zamišljati kao relativne topologije Cantorove kocke, a za pridružene prostore vrijedi da imaju malu induktivnu dimenziju 0 i da su potpuno nepovezani. Vrijedi da su \(\mathcal{T}_{card}\), \(\mathcal{T}^\ast_{card}\) finije od \(\mathcal{T}_{ind}\), \(\mathcal{T}^\ast_{ind}\) , redom, a protuprimjerima pokazujemo da su spomenute topologije u odgovarajućim parovima općenito različite. Također, dokazujemo da se odabirom inverznog niza u kodomeni topologije u parovima poklapaju, te da ih možemo metrizirati konkretnim potpunim ultrametrikama. Prethodne topologije su nam bitne jer je \(k\)-dimenzionalna grupa oblika i gruboga oblika nekog punktiranog prostora \((X, x_0)\) u prirodnoj bijektivnoj vezi redom sa skupom \(pro-\) odnosno \(pro^\ast-\) morfizama između punktirane \(k\)-dimenzionalne sfere i neke \(HPol_0\)-ekspanzije od \((X, x_0)\). Nadalje, budući da pokazujemo da topologije \(\mathcal{T}_{ind}\) i \(\mathcal{T}^\ast_{ind}\), za razliku od \(\mathcal{T}_{card}\)i \(\mathcal{T}^\ast_{card}\), nisu osjetljive na izomorfne transformacije kodomene, to preko netom spomenutih bijektivnih korespondencija možemo koristeći topologije \(\mathcal{T}_{ind}\) i \(\mathcal{T}^\ast_{ind}\) topologizirati grupe oblika i gruboga oblika, redom, neovisno o izboru \(HPol_0\)-ekspanzije. U drugom dijelu radu prvo ćemo se upoznati s već spomenutim topološkim grupama oblika, gdje se topologija kojom su topologizirane u [26] poklapa sa \(\mathcal{T}_{ind}\). Navest ćemo njihova osnovna svojstva, poopćiti neke već poznate primjere, te pružiti eksplicitnu metriku za topološke grupe oblika kompaktnih metričkih prostora. U posljednjem poglavlju ćemo proučavati topologiziranu grupu gruboga oblika nekog punktiranog prostora \((X, x_0)\) i neke dimenzije \(k\). Pokazat ćemo da je ona uistinu topološka grupa, prirodno ćemo je nazvati \(k\)-dimenzionalna topološka grupa gruboga oblika od \((X, x_0)\) i označiti sa \(\check{\pi}^{\ast top}_k(X, x_0)\), te konstrukcijom pripadnog funktora pokazati da je nova invarijanta gruboga oblika. Njezina topološka svojstva bit će naslijeđena od \(\mathcal{T}^\ast_{ind}\), pa će primjerice odmah izaći potpuna regularnost. Povezat ćemo slučaj oblika i gruboga oblika tako da ćemo pokazati da \(\check{\pi}^{\ast top}_k(X, x_0)\) sadrži odgovarajuću topološku grupu oblika kao svoju zatvorenu podgrupu. Također, promatrat ćemo odnos \(\check{\pi}^{\ast top}_k(X, x_0)\) s topološkom grupom gruboga oblika nekog retrakta od \(X\), a i vidjeti ćemo u kojim slučajevima je izbor bazne točke nebitan. Među dokazanim tvrdnjama posebno ističemo teorem o neprekidnosti topoloških grupa gruboga oblika, kao poopćenje njegove nedavno pokazane algebarske varijante. Naposljetku, primjenjujemo dobivene rezultate za konstrukciju zanimljivih primjera, među kojima je i primjer prostora koji nije stabilan, a ima diskretnu, netrivijalnu topološku grupu gruboga oblika.Teorija (topološkog) oblika (vidi [23]) je dobro poznata grana topologije koja pruža općenitiji pogled na topološke prostore (kraće prostore) nego homotopska teorija, a svoje poopćenje ima u teoriji (topološkog) gruboga oblika (vidi [17]). Važne invarijante tih teorija, koje se pridružuju (punktiranim) prostorima su grupe oblika i gruboga oblika, redom. Shape theory (see [23]) is the well known branch of topology which provides more generalised view on topological spaces (shortly spaces) than homotopic theory and it has its own generalisation in the coarse shape theory (see [17]). Important invariants of these theories, which are associated to (pointed) spaces, are shape and coarse shape groups, respectively. Recently, shape group has been topologized in [26], where topological group has been obtained and on similar way we will enrich structure of coarse shape group. In the first part we study different (nontrivial) topologies on the sets of morphisms in categories \(inv\mathcal{-C}, inv^\ast\mathcal{-C}\), where \(\mathcal{C}\) is any category. Then, by using previously mentioned topologies, we topologize in most natural way sets of morphisms in categories \(pro\mathcal{-C}, pro^\ast\mathcal{-C}\) between two fixed objects (i.e. inverse systems in \(\mathcal{C}\)), more precise, in both cases we introduce two topologies, which we denote by \(\mathcal{T}_{ind}\) and \(\mathcal{T}_{card}\) in \(pro-\) case and by \(\mathcal{T}^\ast_{ind}\) and \(\mathcal{T}^\ast_{card}\) in \(pro\ast-\)case. We show that all four mentioned topologies have bases which are consisted of clopen sets and that we may consider them as relative topologies of Cantor cube, and also that associated spaces have small inductive dimension 0 and are completely disconnected. It holds that \(\mathcal{T}_{card}\), \(\mathcal{T}^\ast_{card}\) are finer than \(\mathcal{T}_{ind}\), \(\mathcal{T}^\ast_{ind}\) , respectively, and by using some counterexamples it is shown that those topologies generally di¤er in pairs. Also, we prove that for any inverse sequence in codomain the topologies match in pairs and that in that case we can metrizise them with explicit complete ultrametrics. Aforementioned topologies are important for us because \(k\)-dimensional shape and coarse shape group of some pointed space \((X, x_0)\) is in natural bijective correspondence with set of \(pro-\) and \(pro^\ast-\) morphisms, respectively, between pointed \(k\)-dimensional sphere and \(HPol_0\)-expansion of \((X, x_0)\). Further, since we prove that topologies \(\mathcal{T}_{ind}\) and \(\mathcal{T}^\ast_{ind}\) are not sensitive on isomorphic transformations of codomain, unlike \(\mathcal{T}_{card}\) and \(\mathcal{T}^\ast_{card}\), we can, by using just mentioned bijective correspondences and topologies \(\mathcal{T}_{ind}\) and \(\mathcal{T}^\ast_{ind}\) , topologise shape and coarse shape groups, respectively, independently of choice of \(HPol_0\)-expansion. In the second part we consider mentioned topological shape groups, where their topology in [26] coincides with \(\mathcal{T}_{ind}\). Their main properties are listed, some of known examples are generalised, and also explicit metric for topological shape groups of compact metric spaces is provided. In the last chapter topologised coarse shape group of some pointed space \((X, x_0)\) and some dimension \(k\) is studied. We show that it is truly topological group, which we naturally call \(k\)-dimensional topological coarse shape group of \((X, x_0)\) and denote by \(\check{\pi}^{\ast top}_k(X, x_0)\). By constructing associated functor it is shown that it becomes new coarse shape invariant. Their topological properties are inherited from \(\mathcal{T}^\ast_{ind}\)), so for example it immediately follows that \(\check{\pi}^{\ast top}_k(X, x_0)\) is completely regular. We relate shape and coarse shape case by proving that \(\check{\pi}^{\ast top}_k(X, x_0)\) contains appropriate topological shape group as its closed subgroup. Also, we consider relation of \(\check{\pi}^{\ast top}_k(X, x_0)\) with topological coarse shape group of some retract of \(X\), and we see in which cases specifying basepoint is irrelevant. Among proven facts we highlight continuity of topological coarse shape groups theorem, as generalisation of its recently shown algebraic variant. Finally, our results are applied for constructing interesting examples, among which the example of the space which is not stable, but has discrete, nontrivial topological coarse shape group is most interesting