Homotopy of Linearly Ordered Split–Join Chains in Covering Spaces of Foliated n-Manifold Charts

Abstract

Topological spaces can be induced by various algebraic ordering relations such as, linear, partial and the inclusion-ordering of open sets forming chains and chain complexes. In general, the classifications of covering spaces are made by using fundamental groups and lifting. However, the Riesz ordered n-spaces and Urysohn interpretations of real-valued continuous functions as ordered chains provide new perspectives. This paper proposes the formulation of covering spaces of n-space charts of a foliated n-manifold containing linearly ordered chains, where the chains do not form topologically separated components within a covering section. The chained subspaces within covering spaces are subjected to algebraic split–join operations under a bijective function within chain-subspaces to form simply directed chains and twisted chains. The resulting sets of chains form simply directed chain-paths and oriented chain-paths under the homotopy path-products involving the bijective function. It is shown that the resulting embedding of any chain in a leaf of foliated n-manifold is homogeneous and unique. The finite measures of topological subspaces containing homotopies of chain-paths in covering spaces generate multiplicative and cyclic group varieties of different orders depending upon the types of measures. As a distinction, the proposed homotopies of chain-paths in covering spaces and the homogeneous chain embedding in a foliated n-manifold do not consider the formation of circular nerves and the Nachbin topological preordering, thereby avoiding symmetry/asymmetry conditions.