A new homology theory of orbifolds from triangulations

Abstract

We introduce a new homology theory of compact orbifolds called stratified simplicial homology (or st-homology for short) from some special kind of triangulations adapted to the orbifolds. In the definition of st-homology, the orders of the local groups of the points in an orbifold is encoded in the boundary map so that the theory can capture some structural information of the orbifold. We can prove that st-homology is an invariant under orbifold isomorhpisms and more generally under homotopy equivalences that preserve the orders of the local groups of all the strata. It turns out that the free part of st-homology of an orbifold can be interpreted by the usual simplicial homology of the orbifold and its singular set. So it is the torsion part of st-homology that can really give us new information of an orbifold. In general, the size of the torsion in the st-homology group of a compact orbifold is a nonlinear function on the orders of the local groups of the singular points which may reflect the complexity of the orbifold structure. Moreover, we introduce a wider class of objects called pseudo-orbifolds and develop the whole theory of st-homology in this setting.