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. In a previous paper, the authors 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 $K$.

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 call $K$ a seed if $K$ cannot be obtained by wedgings. First, we show that for a fixed positive integer $\ell $, there are at most finitely many seeds of Picard number $\ell $ supporting characteristic maps. As a corollary, the conjecture proposed by V. V. Batyrev in is solved affirmatively.

Secondly, we investigate a systematicmethod 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.