Extension of 2-Dimensional Planar Systems from Homological Algebra Perspective

Lewis Brew, Joseph Acquah, Newton Amegbey

Abstract


This paper presents 2-Dimensional (2D) planar system together with its changes in a topological space X as a dynamical system. Continuity which is one of the topological properties can be observed from the extension of systems. This paper therefore searches for something computable or an algebraic invariant to identify this topological property of 2D planar system.  The approach is based on the notion of chains and cochains from homological algebra since the vertices and the edges are represented by the chain groups. The connectivity between different parallel chain complexes of the system is represented by the cochain groups. The extension of the 2-Dimensional planar system given by the homomorphism  for each parameter is the sequence of the cochain groups. The surface area of the system is represented by a cocycle and each cocycle is provided by a change in the parameter. The dynamical properties of the system are studied by analysing different cocycle over it. The novel feature is the extension of the systems using the map, the cocycle and the 1st-Cohomology group which are detailed in the paper. 


Keywords


Cochain groups; Chain groups; Cocycle; 1st-Cohomology group; Homotopy, Dynamical System; Exact sequence; Long Exact Sequence; 2-Dimensional (2D) planar system in the space X ; Chain complex and Cochain complex.

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References


. Kinsey, L. C. (1993), Topology of Surfaces, Springer-Verlag N.Y. 271 pp.

. Massey, W. (2000), A Basic Course in Algebraic Topology, Springer-Verlag, New York, 428 pp.

. Weibil, C. A. (1994), An Introduction to homological to Algebra, Cambridge Studies in Advanced Mathematics, 38. Cambridge University Press, Cambridge, 450 pp.


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