The t-singular homology of orbifolds

Abstract

For an orbifold M we define a homology group, called t-singular homology group t-H_q(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-H_q(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 \pi_q(M,x_0) to t-H_q(M) by using of t-modifications. Especially, t-H_1(M) is isomorphic to the abelianization of \pi_1(M,x_0) if M is arcwise connected.