{\displaystyle 1\to \operatorname {Homeo} _{0}(\mathbf {T} ^{n})\to \operatorname {Homeo} (\mathbf {T} ^{n})\to \operatorname {MCG} _{\operatorname {TOP} }(\mathbf {T} ^{n})\to 1.} The mapping class group of higher genus surfaces is much more complicated, and an area of active research. Coloring a torus
The toruss chromatic number is seven, meaning every graph that can be embedded on the torus has a chromatic number of at most seven.
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.