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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Published in: | Afrika mathematica Vol. 34; no. 2 |
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Main Authors: | , |
Format: | Journal Article |
Language: | English |
Published: |
Berlin/Heidelberg
Springer Berlin Heidelberg
01-06-2023
Springer Nature B.V |
Subjects: | |
Online Access: | Get full text |
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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). |
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ISSN: | 1012-9405 2190-7668 |
DOI: | 10.1007/s13370-023-01067-3 |