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 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 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).