Homology Theory on Causal Random Groups

Abstract

The objective of this paper is to analyze a new approach to homology theory that deals with Causal Random Groups, (CRG). It is shown that the evolution of (CRG) can be tracked down as these groups stay within the general random groups category. Random groups are algebraic groups that are not systematically the result of interaction of several random sub algebraic groups. Therefore, in general random groups are unstable. Evolution is change from a known state that can be traced back to that state. If change can not be traced back to its’ initial state, then this change results in a shift to an unknown zone. Change within the unknown zone is designated as evolution in complex domain. The method of tracing in complex domain is outlined and analyzed. It is proven that all change in complex domain stays within the limits defined by causal random algebraic groups. Eventually, (CRG) can reach stability (equilibrium) when repeated change in the complex domain finally leads to emergence from the unknown zone onto a state in real space. A rectangular slab is used to represent causal elements. The relationship between causal elements is depicted as a network of links and paths on each face of the slab, and the space between any two opposing faces. This space is refereed to as internal networks. Hyperplanes and hyperplane pencils are used to cut segments on the networks on the faces, and geodesics are used to explore zones in the internal networks. Zones on any face of the slab are considered to be in the real space, and zones in the internal network are in the complex space. If the boundary points of the zones are on tangent bundles to hyperplane pencils, then these points are non-singular, otherwise the zone contains singularity. Then the focus is shifted onto the internal networks, similar search is done this time with geodesics. Path on non-singular points connect stable, irreducible, cuasal elements, and any sub-group of (CRG) built on these points is complete and optimal.