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# Wedge Operations and Torus Symmetries II

Published:2016-11-29
Printed: Aug 2017
• Suyoung Choi,
Department of Mathematics, Ajou University, 206, World cup-ro, Yeongtong-gu, Suwon, 443-749, Republic of Korea
• Hanchul Park,
School of Mathematics, Korea Institute for Advanced Study (KIAS), 85 Hoegiro Dongdaemun-gu, Seoul 130-722, Republic of Korea
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## 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 $\operatorname{Pic}(K)$ of $K$ is $m-n$. We refer to $K$ as 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 1991 is solved affirmatively. Second, we investigate a systematic method 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.
 Keywords: puzzle, toric variety, simplicial wedge, characteristic map
 MSC Classifications: 57S25 - Groups acting on specific manifolds 14M25 - Toric varieties, Newton polyhedra [See also 52B20] 52B11 - $n$-dimensional polytopes 13F55 - Stanley-Reisner face rings; simplicial complexes [See also 55U10] 18A10 - Graphs, diagram schemes, precategories [See especially 20L05]

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