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
Simplicial sphere "i"-faces of any simplicial "d"-sphere with "f" = "n" vertices. This conjecture was proved for polytopal spheres by Peter McMullen in 1970 and by Richard Stanley for general simplicial spheres in 1975. The g"-conjecture, formulated by McMullen in 1970, asks for a complete characterization of "f"-vectors of simplicial "d"-spheres. In other words, what are the possible sequences of numbers of faces of each dimension for a simplicial "d"-sphere? In the case of polytopal spheres, the answer is given by the g"-theorem, proved in 1979 by Billera and Lee (existence) and Stanley (necessity). It has been conjectured that the same conditions