On quasitoric orbifolds.

*(English)*Zbl 1219.57023As described in this article’s thoroughly and clearly written introduction (Section 1), a quasitoric manifold is a \(2n\)-dimensional manifold \(X\) together with a locally standard action of the \(n\)-torus \(T^n\) so that the quotient space has the structure of an \(n\)-dimensional polytope. Here locally standard means that, given an automorphism \(\theta\) of \(T^n\), the action is locally \(\theta\)-equivariantly diffeomorphic to the standard action of \(T^n\) on \(\mathbb{C}^n\).

The article generalizes this definition by giving two equivalent definitions of a quasitoric orbifold. These equivalent definitions generalize an earlier characterization of quasitoric orbifolds given in M. W. Davis and T. Januszkiewicz [Duke Math. J. 62, No. 2, 417–451 (1991; Zbl 0733.52006)], and are less general than the notion of a torus orbifold defined in A. Hattori and M. Masuda [Osaka J. Math. 40, No. 1, 1–68 (2003; Zbl 1034.57031)]. The first of the two definitions given in this article modifies a known combinatorial model for quasitoric manifolds involving the quotient of a trivial torus bundle \(P^n \times T^n\) on a polytope \(P^n\) by torus subgroups acting on the fibers over the faces of \(P^n\). The second definition generalizes the one given in the preceding paragraph.

The article goes on to determine the orbifold fundamental group and universal cover of a given quasitoric orbifold. It is found that the orbifold fundamental group of a quasitoric orbifold is always of finite order. After adapting the definition of a CW complex to the orbifold setting, the homology of a given quasitoric orbifold with coefficients in \(\mathbb{Q}\) is computed. The rational cohomology ring of a quasitoric orbifold is also computed and related to the Stanley-Reisner face ring of the orbifold’s base polytope. The final section of the article shows that a quasitoric orbifold possesses a stable almost-complex structure. The associated Chen-Ruan cohomology groups are discussed.

The article generalizes this definition by giving two equivalent definitions of a quasitoric orbifold. These equivalent definitions generalize an earlier characterization of quasitoric orbifolds given in M. W. Davis and T. Januszkiewicz [Duke Math. J. 62, No. 2, 417–451 (1991; Zbl 0733.52006)], and are less general than the notion of a torus orbifold defined in A. Hattori and M. Masuda [Osaka J. Math. 40, No. 1, 1–68 (2003; Zbl 1034.57031)]. The first of the two definitions given in this article modifies a known combinatorial model for quasitoric manifolds involving the quotient of a trivial torus bundle \(P^n \times T^n\) on a polytope \(P^n\) by torus subgroups acting on the fibers over the faces of \(P^n\). The second definition generalizes the one given in the preceding paragraph.

The article goes on to determine the orbifold fundamental group and universal cover of a given quasitoric orbifold. It is found that the orbifold fundamental group of a quasitoric orbifold is always of finite order. After adapting the definition of a CW complex to the orbifold setting, the homology of a given quasitoric orbifold with coefficients in \(\mathbb{Q}\) is computed. The rational cohomology ring of a quasitoric orbifold is also computed and related to the Stanley-Reisner face ring of the orbifold’s base polytope. The final section of the article shows that a quasitoric orbifold possesses a stable almost-complex structure. The associated Chen-Ruan cohomology groups are discussed.

Reviewer: Elizabeth Stanhope (Portland)

##### MSC:

57R19 | Algebraic topology on manifolds and differential topology |

57R91 | Equivariant algebraic topology of manifolds |

57R18 | Topology and geometry of orbifolds |

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\textit{M. Poddar} and \textit{S. Sarkar}, Osaka J. Math. 47, No. 4, 1055--1076 (2010; Zbl 1219.57023)

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