Wedge operations and torus symmetries II

Abstract

A fundamental idea in toric topology is that classes of manifolds with well-behaved torus actions (simply, toric spaces) are classified by pairs of simplicial complexes and (non-singular) characteristic maps. The authors in their previous paper provided a new way to find all characteristic maps on a simplicial complex K(J) obtainable by a sequence of wedgings from K. The main idea was that characteristic maps on K theoretically determine all possible characteristic maps on a wedge of K. In this work, we further develop our previous work for classification of toric spaces. For a star-shaped simplicial sphere K of dimension n-1 with m vertices, the Picard number Pic(K) of K is m-n. We refer to K a seed if K cannot be obtained by wedgings. First, we show that, for a fixed positive integer , there are at most finitely many seeds of Picard number supporting characteristic maps. As a corollary, the conjecture proposed by V. V. Batyrev in 1991 is solved affirmatively. Second, we investigate a systematic way to find all characteristic maps on K(J) using combinatorial objects called (realizable) puzzles that only depend on a seed K. These two facts lead to a practical way to classify the toric spaces of fixed Picard number.