, where one substitutes for Aut the appropriate group for the category to which X belongs. Here denotes the 0-th homotopy group of a space. So in general, there is a short exact sequence of groups: Frequently this sequence is not split. If working in the homotopy category, the mapping class group of X is the group of homotopy classes of homotopy equivalences of X. There are many subgroups of mapping class groups that are frequently studied. If M is an oriented manifold, would be the orientation-preserving automorphisms of M and so the mapping class group of M (as an oriented manifold) would be index two in the mapping class group of M (as an unoriented manifold) provided M admits an orientation-reversing automorphism. Similarly, the subgroup that acts as the identity on all the homology groups of M is called the Torelli group of M. Examples Sphere In any category (smooth, PL, topological, homotopy) corresponding to maps of degree ±1. Torus In the homotopy category This is because the n-dimensional torus is an Eilenberg–MacLane space. For other categories if , one has the following split-exact sequences: In the category of topological spaces In the PL-category (⊕ representing direct sum). In the smooth category where are the Kervaire–Milnor finite abelian groups of homotopy spheres and is the group of order 2. Surfaces The mapping class groups of surfaces have been heavily studied, and are sometimes called Teichmüller modular groups (note the special case of above), since they act on Teichmüller space and the quotient is the moduli space of Riemann surfaces homeomorphic to the surface. These groups exhibit features similar both to hyperbolic groups and to higher rank linear groups. They have many applications in Thurston's theory of geometric three-manifolds (for example, to surface bundles). The elements of this group have also been studied by themselves: an important result is the Nielsen–Thurston classification theorem, and a generating family for the group is given by Dehn twists which are in a sense the "simplest" mapping classes. Every finite group is a subgroup of the mapping class group of a closed, orientable surface,; in fact one can realize any finite group as the group of isometries of some compact Riemann surface (which immediately implies that it injects in the mapping class group of the underlying topological surface). Non-orientable surfaces Some non-orientable surfaces have mapping class groups with simple presentations. For example, every homeomorphism