We study the number of D-J classes on P L-spheres with a few vertices. The Picard number of a P L-sphere with m vertices of dimension n − 1 is defined as m − n. It is known that all PL-spheres with Picard numbers less than 4 are polytopal, and the number is well-known by Perles. Furthermore, the numbers of D-J classes over such PL-spheres have been computed by Choi and Park. Nevertheless, only little is known about P L-spheres with the Picard number 4.
In this thesis, we focus on PL-spheres of the Picard number 4. It is known that there are 5 PL-spheres of dimension 2 as a corollary of Steinitz’s theorem, and 39 of dimension 3 by Barnette. We construct two algorithms to count 4 dimensional P L-spheres of the Picard number 4; one gives the lower bound and the other gives the upper bound. Using this, we show that there are exactly 337 P L-spheres of dimension 4 with 9 vertices. As a corollary, we show that there are exactly 15757 D-J classes over such PL-spheres.
