Rad HAZU, Matematičke znanosti, Vol. 21 (2017), 205-217.

TOPOLOGICAL COARSE SHAPE GROUPS OF COMPACT METRIC SPACES

Nikola Koceić Bilan and Zdravko Čuka

Faculty of science, University of Split, Ruđera Boškovića 33, 21000 Split, Croatia
e-mail: koceic@pmfst.hr

Faculty of civil engineering, architecture and geodesy, University of Split, 21000 Split, Croatia


Abstract.   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 πk*d*(X,x0). We have proven some important properties of πk*d*(X,x0) and provided few interesting examples.

2010 Mathematics Subject Classification.   55P55, 55Q05, 55N99.

Key words and phrases.   Inverse system, pro-category, pro*-category, expansion, shape, coarse shape, homotopy pro-group, shape group, coarse shape group, topological shape homotopy group, coarse shape path connectedness.


Full text (PDF) (free access)

DOI: http://doi.org/10.21857/9xn31cvr1y


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