3-sphere two 3-balls. This construction is analogous to a construction of a 2-sphere, performed by gluing the boundaries of a pair of disks. A disk is a 2-ball, and the boundary of a disk is a circle (a 1-sphere). Let a pair of disks be of the same diameter. Superpose them and glue corresponding points on their boundaries. Again one may think of the third dimension as temperature. Likewise, we may inflate the 2-sphere, moving the pair of disks to become the northern and southern hemispheres. After removing a single point from the 2-sphere, what remains is homeomorphic to the Euclidean
Exotic sphere Similar manifolds are called Brieskorn spheres. Given an (orientation-preserving) diffeomorphism , gluing the boundaries of two copies of the standard disk "D" together by "f" yields a manifold called a "twisted sphere" (with "twist" "f"). It is homotopy equivalent to the standard "n"-sphere because the gluing map is homotopic to the identity (being an orientation-preserving diffeomorphism, hence degree 1), but not in general diffeomorphic to the standard sphere. Setting formula_25 to be the group of twisted "n"-spheres (under connect sum), one obtains the exact sequence For , every exotic "n"-sphere is diffeomorphic to a twisted sphere, a result proven by