The T-singular Homology of Orbifolds

For an orbifold M we define a homology group, called t-singular homology group t-Hq(M), which depends not only on the topological structure of the underlying space of M , but also on the orbifold structure of M . We prove that it is a b-homotopy invariant of orbifolds. If M is a manifold, t-Hq(M) coincides with the usual singular homology group. We calculate the t-singular homology groups with Z-coefficients of several orbifolds. The t-singular homology group with rational coefficients of an orbifold M is isomorphic to the singular homology group with rational coefficients of M (and to that of |M |). For each q we define a homomorphism from the orbifold homotopy group πq(M,x0) to t-Hq(M) by using of t-modifications. Especially, t-H1(M) is isomorphic to the abelianization of π1(M,x0) if M is arcwise connected.