Extension of 2Dimensional Planar Systems from Homological Algebra Perspective
Abstract
This paper presents 2Dimensional (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 2Dimensional 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 1^{st}Cohomology group which are detailed in the paper.
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. Kinsey, L. C. (1993), Topology of Surfaces, SpringerVerlag N.Y. 271 pp.
. Massey, W. (2000), A Basic Course in Algebraic Topology, SpringerVerlag, 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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