For any number of holes, the formula generalizes to V − E + F = 2 − 2N, where N is the number of holes. The term "toroidal polyhedron" is also used for higher-genus polyhedra and for immersions of toroidal polyhedra. Automorphisms
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 surjective onto the group GL(n, Z) of invertible integer matrices, which can be realized as linear maps on the universal covering space Rn that preserve the standard lattice Zn (this corresponds to integer coefficients) and thus descend to the quotient.
{\displaystyle \operatorname {MCG} _{\operatorname {Ho} }(\mathbf {T} ^{n})=\operatorname {Aut} (\pi _{1}(X))=\operatorname {Aut} (\mathbf {Z} ^{n})=\operatorname {GL} (n,\mathbf {Z} ).} Since the torus is an Eilenberg–MacLane space K(G, 1), its homotopy equivalences, up to homotopy, can be identified with automorphisms of the fundamental group); all homotopy equivalences of the torus can be realized by homeomorphisms – every homotopy equivalence is homotopic to a homeomorphism.