Editing Mapping class group (section)

=== Torus ===
In the [[homotopy category]]
:<math> \operatorname{MCG}(\mathbf{T}^n) \simeq \operatorname{GL}(n,\Z). </math>
This is because the [[Torus#n-dimensional torus|n-dimensional torus]] <math>\mathbf{T}^n = (S^1)^n</math> is an [[Eilenberg–MacLane space]].

For other categories if <math>n\ge  5</math>,<ref>{{cite book |first=A.E. |last=Hatcher |chapter=Concordance spaces, higher simple-homotopy theory, and applications |chapter-url={{GBurl|6hsDCAAAQBAJ|p=3}} |title=Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), Part 1 |series=Proceedings of Symposia in Pure Mathematics |publisher= |location= |date=1978 |volume=32 |issue=1 |isbn=978-0-8218-9320-3 |pages=3–21 |doi=10.1090/pspum/032.1/520490 |mr=0520490}}</ref> one has the following split-exact sequences:

In the [[category of topological spaces]]
:<math>0\to \Z_2^\infty\to \operatorname{MCG}(\mathbf{T}^n) \to \operatorname{GL}(n,\Z)\to 0</math>
In the [[Piecewise linear manifold|PL-category]]
:<math>0\to \Z_2^\infty\oplus\binom n2\Z_2\to \operatorname{MCG}(\mathbf{T}^n)\to \operatorname{GL}(n,\Z)\to 0</math>
(⊕ representing [[direct sum]]).
In the [[Smooth manifold|smooth category]]
:<math>0\to \Z_2^\infty\oplus\binom n2\Z_2\oplus\sum_{i=0}^n\binom n i\Gamma_{i+1}\to \operatorname{MCG}(\mathbf{T}^n)\to \operatorname{GL}(n,\Z)\to 0</math>

where <math>\Gamma_i</math> are the Kervaire–Milnor finite abelian groups of [[homotopy sphere]]s and <math>\Z_2</math> is the group of order 2.