 The homeomorphism group (or the subgroup of diffeomorphisms) of the torus is studied in geometric topology.Its mapping class group ( the connected components of the homeomorphism group) is isomorphic to the group GL("n", Z) of invertible integer matrices, and can be realized as linear maps on the universal covering space R"n" that preserve the standard lattice Z"n" (this corresponds to integer coefficients) and thus descend to the quotient.At the level of homotopy and homology, the mapping class group can be identified as the action on the first homology (or equivalently, first cohomology, or on the fundamental group, as these are all naturally isomorphic; also the first cohomology group generates the cohomology algebra