Combinatorics of Constructible Complexes

Preface Everything started from one book. I happened to buy the textbook \Lectures on Poly-topes" 98] written by Prof. G unter M. Ziegler, at the university bookstore about ve years ago. I bought it only because the gures (especially of permutahedra and of zonotopal tilings) interested me, but the book turned out to be a very good introduction to the world of poly-topes, starting from fundamentals and containing many recent results. Among the many topics, especially Lecture 8 on shellability attracted my interest. Because the shellability is a concept which formalizes a very natural construction of objects by adding cells (= facets) one by one, it is conceivable that triangulations and polytopal decompositions of balls and spheres are shellable. But surprisingly, many counterexamples, that is, non-shellable decompositions of balls and spheres are known. One of such examples, Danzer's cube, is described in the book. I read that part repeatedly and spent much time imaging what is happening on the ball, and then proceeded to other non-shellable balls according to the references in the book. Still the diierence between shellable decompositions and non-shellable decompositions was a big mystery to me, and I have been thinking of this for years. Soon I decided shellability should be the theme of my doctoral study. What I had in my mind at that time was to give some characterizations of non-shellability, though this aim has not been achieved yet. During the study, I fell to thinking that why shellings can add only one facet in one step: what will happen if we allow a lump of facets to be added at each step? After formulating this \generalized" deenition of shelling, I thought that I had seen the same formulation somewhere before. I was right. It is given in the book \Cohen-Macaulay Rings" by Profs. Bruns and Herzog 26], named \constructibility." (This concept turned out to go back to a 1972 paper of Hochster 49].) From this point my main interest shifted to constructibility. Very few works, however, have been done about constructibility. All I could nd were a few papers, each of which made a few statements about constructibility, so I decided to study constructibility myself. I started with the problem whether or not there are non-constructible triangulations of 3-balls or spheres analogous to the case of shellability. My rst attempt was to show that every triangulated 3-ball is constructible, which failed as is observed …