On the Foundation of Algebraic Topology.
G. Fors
posted in Mathematics on Monday, November 2nd, 2009
In the 70:th the combinatorialists begun to systematically relate simplicial complexes and polynomial algebras (called Stanley-Reisner rings). The simplex and the join definitions were modified, turning the empty simplicial complex into a zero object w.r.t. to simplicial join and thereby losing its former role as join-unit - a role taken over by a new (-1)-dimensional simplicial complex containing only the empty simplex. </p> <p>There can be no realization functor targeting the classical category of topological spaces that turns the contemporary simplicial join into topological join unless a (-1)-dimensional space is introduced as a topological join-unit. This is done in this paper and it enables an augmental homology theory that unifies the classical relative and reduced homology functors and allows a Kynneth Theorem for simplicial resp. topological pair-joins.
