Blowdown, \(k\)-wedge and evenness of quasitoric orbifolds (Q6558883)

The wedge construction introduced by \textit{G. Ewald} (see [Discrete Comput. Geom. 1, 115--122 (1986: Zbl 0597.52009)]) and the more general \(J\)-construction introduced by \textit{A. Bahri} et al. [Adv. Math. 225, No. 3, 1634--1668 (2010; Zbl 1197.13021)] are operations to construct new simplicial complexes from a given simplicial complex. Since a simplicial complex with some additional data corresponds to an object (manifold or orbifold) with a (real or compact) torus action, these operations are studied in toric geometry and topology. For example, they are used to construct interesting toric objects from given toric objects (see [\textit{S. Choi} and \textit{H. Park}, Forum Math. 29, No. 3, 543--553 (2017; Zbl 1377.57022)]).\N\NIn the paper under review, the authors introduce the polytopal \(k\)-wedge construction and the blowdown of a simple polytope which may be regarded as a generalization of the \(J\)-construction. They study the properties of these constructions and their applications to the topology of quasitoric orbifolds, i.e., orbifolds with a torus action whose orbit space has the structure of a simple convex polytope. Note that the authors prove that the polytopal \(k\)-wedge is the blowdown of \(Q\times \Delta^{k}\) along the facet \(F\times \Delta^{k}\) in Corollary 4.8.\N\NThe central technical result of the paper is Theorem 4.9, where the authors show that there exists a retraction sequence for \(Q'\), induced from that of \(Q\). The retraction sequence is a concept introduced by \textit{A. Bahri} et al. [Algebr. Geom. Topol. 17, No. 6, 3779--3810 (2017; Zbl 1386.14187)]. Using Theorem 4.9, the authors prove in Theorem 5.9 that if \(Q'\) is obtained by the blowdown of \(Q\) (combinatorially), and if there exists a quasitoric orbifold \(X(Q,\lambda)\) and a prime number \(p\) satisfying certain conditions, then there exists a quasitoric orbifold \(X(Q',\lambda')\) which can be obtained by the blowdown of \(X(Q,\lambda)\).\N\NAs an application of Theorem 5.9, the authors establish the evenness condition (i.e., its cohomology has no torsion and is concentrated in even degree) for quasitoric orbifolds in Corollary 5.14. In the final section, they construct an example of the blowdown of a quasitoric orbifold that cannot be obtained via the \(J\)-construct (see Example 6.7).