 It follows that the Euler characteristic of the "n"-torus is 0 for all "n".The cohomology ring "H"•(T"n", Z) can be identified with the exterior algebra over the Z-module Z"n" whose generators are the duals of the "n" nontrivial cycles.As the "n"-torus is the "n"-fold product of the circle, the "n"-torus is the configuration space of "n" ordered, not necessarily distinct points on the circle.