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: https://doi.org/10.21857/9xn31cvr1y

References:

  1. N. Koceić Bilan, The coarse shape groups, Topology Appl. 157 (2010) 894-901.
    MathSciNet     CrossRef

  2. N. Koceić Bilan, Computing coarse shape groups of solenoids, Math. Commun. 14 (2014), 243-251.
    MathSciNet

  3. N. Koceić Bilan and N. Uglešić, The coarse shape path connectedness, Glas. Mat. Ser. III 46(66) (2011), 491-505.
    MathSciNet     CrossRef

  4. N. Koceić Bilan and N. Uglešić, The coarse shape, Glas. Mat. Ser. III 42(62) (2007), 145-187.
    MathSciNet     CrossRef

  5. T. Nasri, F. Ghanei, B. Mashayekhy and H. Mirebrahimi, On topological shape homotopy groups, Topology Appl. 198 (2016), 22-33.
    MathSciNet     CrossRef

  6. F. Ghanei, H. Mirebrahimi, B. Mashayekhy and T. Nasri, Topological coarse shape homotopy groups, Topology Appl. 219 (2017), 17-28.
    MathSciNet     CrossRef

  7. S. Mardešić and J. Segal, Shape theory, North-Holland, Amsterdam, 1982.
    MathSciNet

  8. N. Uglešić, Metrization of pro-morphism sets, Glas. Mat. Ser. III 44(64) (2009), 499-531.
    MathSciNet     CrossRef

  9. N. Uglešić, Ultrametrization of pro*-morphism sets, Math. Commun. 18 (2013), 19-47.
    MathSciNet

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