Editing Torus (section)

== 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 <math>\operatorname{GL}(n,\mathbf{Z})</math> of invertible integer matrices, which can be realized as linear maps on the universal covering space <math>\mathbf{R}^{n}</math> that preserve the standard lattice <math>\mathbf{Z}^{n}</math> (this corresponds to integer coefficients) and thus descend to the quotient.

At the level of [[homotopy]] and [[homology (mathematics)|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:
: <math>\operatorname{MCG}_{\operatorname{Ho}}(T^n) = \operatorname{Aut}(\pi_1(X)) = \operatorname{Aut}(\mathbf{Z}^n) = \operatorname{GL}(n,\mathbf{Z}).</math>

Since the torus is an [[Eilenberg–MacLane space]] {{math|''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.

Thus the [[short exact sequence]] of the mapping class group splits (an identification of the torus as the quotient of <math>\mathbf{R}^{n}</math> gives a splitting, via the linear maps, as above):
: <math>1 \to \operatorname{Homeo}_0(T^n) \to \operatorname{Homeo}(T^n) \to \operatorname{MCG}_{\operatorname{TOP}}(T^n) \to 1.</math>

The mapping class group of higher genus surfaces is much more complicated, and an area of active research.