2001/09/11 by G. Fors, Göran Fors, Fors, G.
Computer Science · Mathematics · #(Primary) 57P05. 05-XX #55N10 #55Nxx #Algebraic Topology (math.AT) #Combinatorics (math.CO) #Commutative Algebra and Its Applications #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis #math.AT #math.CO #msc:05-XX #msc:55N10 #msc:55Nxx #msc:57P05.
paper · pdf · doi:10.48550/arxiv.math/0109073
These 32 pages has been prepared using AMSTeX with \documentstyle{amsppt} in a MikTeX2.0 environment. It's an improvement based on REPORTS/ Department of Mathematics, University of Stockholm, Sweden; A Homology Theory Based on the Existence of a (-1)-dimensional Simplex; by G. Fors, 1994 - No 3. (25 pages)
arxiv created 2001/09/11 · openalex publication_date 2001/09/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The "simplicial complexes" and "join" (*) today used within combinatorics aren't the classical concepts, cf. Spanier (1966) p. 108-9, but, exept for ∅, complexes having ∅ as a subcomplex resp. Σ1 * Σ2 := σ1 ∪ σ2 | \sigmai ∈ \Sigmai implying a tacit change of unit element w.r.t. the join operation, from ∅ to ∅. 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) ∪ (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.