The t-singular homology of orbifolds

For an orbifold M we define a new homology group, called t-singular homology group t - H q ( M ) by using singular simplicies intersecting ‘transversely’ with ΣM. The rightness of this homology group is ensured by the facts that the 1-dimensional homology group t - H 1 ( M ) is isomorphic to the abelianization of the orbifold fundamental group π 1 ( M , x 0 ) . If M is a manifold, t - H q ( M ) coincides with the usual singular homology group. We prove that it is a ‘b-homotopy’ invariant of orbifolds and develop many algebraic tools for the calculations. Consequently we calculate the t-singular homology groups of several orbifolds.