Augmental Homology and the Kynneth Formula for Joins

Abstract

The "simplicial complexes" and "join" (*) today used within combinatorics aren't the classical concepts, cf. Spanier (1966) p. 108-9, but, exept for \emptyset, complexes having {\emptyset} as a subcomplex resp. \Sigma1 * \Sigma2 := {\sigma1 \cup \sigma2 | \sigmai \in \Sigmai} implying a tacit change of unit element w.r.t. the join operation, from \emptyset to {\emptyset}. Extending the classical realization functor to this category of simplicial complexes we end up with a "restricted" category of topological spaces, "containing" the classical and where the classical (co)homology theory, as well as the ad-hoc invented reduced versions, automatically becomes obsolete, in favor of a unifying and more algebraically efficient theory. This very modest category modification greatly improves the interaction between algebra and topology. E.g. it makes it possible to calculate the homology groups of a topological pair-join, expressed in the relative factor groups, leading up to a truly simple boundary formula for joins of manifolds: Bd(X1 * X2) = ((BdX1 * X2) \cup (X1 * BdX2)), the product counterpart of which is true also classically. It is also easily seen that no finite simplicial n-manifold has an (n-2)-dimensional boundary, cf. Cor. 1 p. 26, and that simplicial homology manifolds with the integers as koefficient module are all locally orientable, cf. Cor. 2 p. 29.