Wedge Operations and Torus Symmetries II

Choi, Suyoung ; Park, Hanchul

Canadian journal of mathematics, Tome 69 (2017) no. 4, pp. 767-789

Résumé

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 inis 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.

Détail

DOI : 10.4153/CJM-2016-037-4

Mots-clés : 57S25, 14M25, 52B11, 13F55, 18A10, puzzle, toric variety, simplicial wedge, characteristic map