D. K. Biss (Topology and its Applications 124 (2002) [355][356][357][358][359][360][361][362][363][364][365][366][367][368][369][370][371] introduced the topological fundamental group and presented some interesting basic properties of the notion. In this article we intend to extend the above notion to homotopy groups and try to prove some similar basic properties of the topological homotopy groups. We also study more on the topology of the topological homotopy groups in order to find necessary and sufficient conditions for which the topology is discrete. Moreover, we show that studying topological homotopy groups may be more useful than topological fundamental groups.2000 Mathematics Subject Classification. 55Q05; 55U40; 54H11; 55P35.

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The article describes a certain computation method of ( , )-arcs to construct the number of distinct ( , 4)-arcs in PG(2,17) for = 7, … ,48. In this method, a new approach employed to compute the number of ( , )-arcs and the number of distinct ( , )-arcs respectively. This approach is based on choosing the number of inequivalent classes { 4 , 3 , 2 , 1 , 0 } of -secant distributions that is the number of 4-secant, 3secant, 2-secant, 1-secant and 0-secant in each process. The maximum size of ( , 4)-arc that has been constructed by this method is = 48. The new method is a new tool to deal with the programming difficulties that sometimes may lead to programming problems represented by the increasing number of arcs. It is essential to reduce the established number of ( , )-arcs in each construction especially for large value of and then reduce the running time of the calculation. Therefore, it allows to decrease the memory storage for the calculation processes. This method's effectiveness evaluation is confirmed by the results of the calculation where a largest size of complete ( , 4)-arc is constructed. This research's calculation results develop the strategy of the computational approaches to investigate big sizes of ( , )-arcs in (2, ) where it put more attention to the study of the number of the inequivalent classes of -secants of ( , )-arcs in (2, ) which is an interesting aspect. Consequently, it can be used to establish a large value of .

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In this paper, the classification of (k;4)-arcs up to projective inequivalence for k < 10 in PG(2,13) is introduced in details according to their inequivalent number, stabilisers, the action of each stabiliser on the associated arc, and the inequivalent classes N c of secant distributions of arcs. Here, the strategy is to start from the projective line PG(1,13) where there are three projectively inequivalent tetrads.