Weighted homology theory of orbifolds and Weighted Polyhedra (2106.06794v4)

Abstract: We introduce two new homology theories of orbifolds from some special type of triangulations adapted to orbifolds, called AW-homology and DW-homology. The main idea in the definitions of these two homology theories is that we use divisibly weighted simplices as building blocks of an orbifold and encode the orders of the local groups of the orbifold in the boundary maps of the chain complexes so that these two theories can reflect some information of the singular points. We prove that AW-homology and DW-homology are invariants of orbifolds under isomorphisms and more generally under certain type of homotopy equivalences of orbifolds. Moreover, we find that there exists a natural graded commutative product in the cohomology theory associated to DW-homology, which generalizes the cup product of the ordinary simplicial cohomology. In addition, we introduce a wider class of objects called weighted polyhedra and develop the whole theory of AW-homology and DW-homology in this wider setting. Our goal is to generalize the whole simplicial homology theory to all triangulizable topological spaces with a weight at each point where the weights are compatible with the triangulations in a certain sense.