Dyck fundamental group on arcwise-connected polygon cycles

This paper introduces the Dyck fundamental group presentation of arcwise-connected polygon cycles resulting from invariant transforms that preserve the ratio of collinear points in the Desargues affine plane. Here, a fundamental group is a collection of path-connected free groups on homotopic path c...

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Bibliographic Details
Published in:Afrika mathematica Vol. 34; no. 2
Main Authors: Peters, James F., Zaka, Orgest
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer Berlin Heidelberg 01-06-2023
Springer Nature B.V
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Summary:This paper introduces the Dyck fundamental group presentation of arcwise-connected polygon cycles resulting from invariant transforms that preserve the ratio of collinear points in the Desargues affine plane. Here, a fundamental group is a collection of path-connected free groups on homotopic path cycles geometrically realized as arcwise-connected polygon cycles. For a J.H.C. Whitehead closure-finite weak (CW) topological space X , a path is a continuous map h : X × I → X over the unit interval I = [ 0 , 1 ] . The geometric realization of a path is an edge in a polygon cycle, which is a sequence of edges attached to each other with no end edge and no self-loops. The main results in this paper are (1) Every collection of arcwise-connected Dyck polygon cycles in the Desargues affine plane is a geometric realization of a collection path-connected homotopic path cycles (Theorem 10). (2) Every collection of arcwise-connected clusters of triangles in the Desargues affine plane is a geometric realization of a Dyck fundamental group (Theorem 11).
ISSN:1012-9405
2190-7668
DOI:10.1007/s13370-023-01067-3