of the real projective plane is isotopic to the identity: The mapping class group of the Klein bottle K is: The four elements are the identity, a Dehn twist on a two-sided curve which does not bound a Möbius strip, the y-homeomorphism of Lickorish, and the product of the twist and the y-homeomorphism. It is a nice exercise to show that the square of the Dehn twist is isotopic to the identity. We also remark that the closed genus three non-orientable surface N3 (the connected sum of three projective planes) has: This is because the surface N has a unique class of one-sided curves such that, when N is cut open along such a curve C, the resulting surface is a torus with a disk removed. As an unoriented surface, its mapping class group is . (Lemma 2.1). 3-Manifolds Mapping class groups of 3-manifolds have received considerable study as well, and are closely related to mapping class groups of 2-manifolds. For example, any finite group can be realized as the mapping class group (and also the isometry group) of a compact hyperbolic 3-manifold. Mapping class groups of pairs Given a pair of spaces (X,A) the mapping class group of the pair is the isotopy-classes of automorphisms of the pair, where an automorphism of (X,A) is defined as an automorphism of X that preserves A, i.e. f: X → X is invertible and f(A) = A. Symmetry group of knot and links If K ⊂ S3 is a knot or a link, the symmetry group of the knot (resp. link) is defined to be the mapping class group of the pair (S3, K). The symmetry group of a hyperbolic knot is known to be dihedral or cyclic, moreover every dihedral and cyclic group can be realized as symmetry groups of knots. The symmetry group of a torus knot is known to be of order two Z2. Torelli group Notice that there is an induced action of the mapping class group on the homology (and cohomology) of the space X. This is because (co)homology is functorial and Homeo0 acts trivially (because all elements are isotopic, hence homotopic to the identity, which acts trivially, and action on (co)homology is invariant under homotopy). The kernel of this action is the Torelli group, named after the Torelli theorem. In the case of orientable surfaces, this is the action on first cohomology H1(Σ) ≅ Z2g. Orientation-preserving