| Sažetak | U ovom radu dajemo novu karakterizaciju grupa gruboga oblika punktiranih topoloških prostora, temeljnih invarijanti teorije gruboga oblika. Uvodimo novi funktor \(\widetilde R\) iz \(pro^{*}\text{-}Grp\) u \( pro\text{-}Grp\) kategoriju. Pokaže se da je \(\left( \widetilde R\circ pro^{*}\text{-}\pi _k \right) \left(X,x_0\right) \) za neki punktirani topološki prostor \((X,x_0)\) upravo \(pro\text{-}\)grupa gruboga oblika \(pro\text{-} \check\pi _k^*\left(X,x_0\right)\). Nadalje, pokaže se da za \(pro\text{-} \check\pi _k^*\equiv \widetilde R\circ pro^*\text{-} \check\pi _k\) vrijedi \(\ds \lim\limits_{\leftarrow}pro\text{-} \check\pi _k^*=\check\pi _k^*\), što znači da je \(pro\text{-} \check\pi _k^*\) puni analogon funktoru \(pro\text{-}\pi_k\). Funktor \(\widetilde{R}\) nam omogućuje i da definiramo homološku grupu gruboga oblika i homološku \(pro\)-grupu gruboga oblika topološkog prostora. Koristeći rezultate iz [5], [9] i [16] u radu istražujemo vezu između \( pro \)-grupa gruboga oblika i homoloških \( pro \)-grupa gruboga oblika u smislu Hurewiczeva teorema, te konačno i vezu između grupa gruboga oblika i homoloških grupa gruboga oblika. Navodimo analogne rezultate i u relativnom slučaju, za punktirane parove topoloških prostora. U radu također proučavamo redukciju kategorije \(pro^*\)-\(Top\) s pomoću novog funktora \(\widetilde R\colon pro^*\)-\(Top \to pro\)-\(Top\) koji \(*\)-morfizme prikazuje kao morfizme \( pro \)-kategorije među inverznim sustavima u kojima su termi reducirane potencije topoloških prostora. S ciljem reduciranja \(pro^*\)-\(HTop\) kategorije, ponudili smo poopćenje pojma homotopije, relaciju koju smo nazvali box-homotopijom. Dokazali smo da je box-homotopija relacija ekvivalencije na \(Top(X,Y)\) i da je dobro usklađena s kompozicijom, što omogućuje konstrukciju nove kvocijentne kategorije. Pokazalo se, međutim, da su sva preslikavanja međusobno box-homotopna, odnosno da je klasifikacija morfizama po relaciji box-homotopnosti trivijalna. |
| Sažetak (engleski) | The coarse shape theory is a relatively new branch of algebraic topology. It was introduced in [11] about ten years ago as a generalization of the shape theory, providing a rougher tool for classifying locally bad topological spaces. In this thesis, we give a new characterization of the coarse shape groups of pointeed topological spaces, the fundamental invariants of this theory. We propose a functor \(\widetilde R\) from \(pro^{*}\text{-}Grp\) to \( pro\text{-}Grp\), which represents morphisms in \(pro^{*}\)-category as morphisms in \(pro\)-category between more complex objects. It turns out that \(\left( \widetilde R\circ pro^{*}\text{-}\pi _k \right) \left(X,x_0\right) \) is a \(pro\text{-}\)coarse shape group \(pro\text{-} \check\pi _k^*\left(X,x_0\right)\). Furthermore, for \(pro\text{-} \check\pi _k^*\equiv \widetilde R\circ pro^*\text{-} \check\pi _k\), the equality \(\ds \lim\limits_{\leftarrow}pro\text{-} \check\pi _k^*=\check\pi _k^*\) holds. Since the shape group functor \(\check\pi _k\) is defined by \(\ds \check\pi _k=\lim\limits_{\leftarrow}pro\text{-} \pi _k\), the functor \(pro\text{-} \check\pi _k^*\) is a full analog of \(pro\text{-}\pi_k\). Naturally, \(pro\text{-} \check\pi _k^*\) gives more informations then \(\check\pi _k^*\) because we lose some information in limit (as well as with \(pro\text{-}\pi_k\) and \(\check{\pi}_k\)). We use this new functor to define coarse shape homology group of a topological space. The Hurewicz theorem, fundamental result of algebraic topology that relates homotopy and homology groups, was established also for \(pro\)-groups as well as for \(pro^*\)-groups, and now we bring its version for \(pro\)-coarse shape groups. This enables us to relate coarse shape groups and coarse shape homology groups. We prove that the first nontrivial coarse shape group and coarse shape homology group of a pointed continuum are isomorphic, assertion that does not hold for shape groups. In this thesis we observe a reduced power \(\widetilde X\) of a topological space \(X\), a product of topological space \(X\) by itself countably many times, given the box topology, and reduced to a quotient space by an equivalence relation saying that two sequences of elements in \(X\) are related if they differ only on finite number of coordinates. We list some of its properties. For instance, it preserves separation properties from the original space. Also, in \(\widetilde X\), intersections of countably many open sets, in other words \(G_{\delta}\) sets, are open. In literature such a space is called a \(P\)-space in the sense of Gillman–Henriksen. This construction enables us to introduce a new functor \(\widetilde R\) from \(pro^{*}\text{-}Top\) to \( pro\text{-}Top\), which represents morphisms in \(pro^{*}\)-category as morphisms in \(pro\)-category between more complex objects. In the sequel we propose a generalisation of the notion of homotopy, a relation that we call box-homotopy and denote by \(\underset{\square}{\sim}\). Naturally, all homotopic maps are box-homotopic, and we provide an example showing that the opposite does not hold. Then we construct a new category, \(\underset{\square}{\sim}\), with topological spaces as objects and box-homotopy classes of continuous maps as morphisms, as it proves out that box-homotopy is an equivalence relation on \(Top(X,Y)\), and that it is well adjusted with the composition. Unfortunately, all the mappings were shown to be box-homotopic, that is, the classification of morphisms by box-homotopy relation is trivial. Nevertheless, we have provided the results of the research because of the interesting constructs and useful inner-conclusions. |
