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 $\mathrm{Pic}(K)$ 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 $\ell$, 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 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.

21 pages, 1 figure; corrected Theorem 2.2 and added Corollary 2.6 to prove a conjecture of Batyrev in 1991 in the second version

Cited by in corpus (4)

• An algorithmic strategy for finding characteristic maps over wedged simplicial complexes
• The characterization of $(n-1)$-spheres with $n+4$ vertices having maximal Buchstaber number
• Stellar subdivisions, wedges and Buchstaber numbers
• Toric wedge induction and toric lifting property for piecewise linear spheres with a few vertices