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On quasitoric orbifolds. (English) Zbl 1219.57023
As 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.

##### MSC:
 57R19 Algebraic topology on manifolds and differential topology 57R91 Equivariant algebraic topology of manifolds 57R18 Topology and geometry of orbifolds
##### Keywords:
quasitoric; orbifold; manifold
##### Citations:
Zbl 0733.52006; Zbl 1034.57031
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##### References:
 [1] A. Adem, J. Leida and Y. Ruan: Orbifolds and Stringy Topology, Cambridge Tracts in Mathematics 171 , Cambridge Univ. Press, Cambridge, 2007. · Zbl 1157.57001 [2] V.M. Buchstaber and N. Ray: Tangential structures on toric manifolds, and connected sums of polytopes , Internat. Math. Res. Notices (2001), 193–219. · Zbl 0996.52013 [3] G.E. Bredon: Introduction to Compact Transformation Groups, Pure and Applied Mathematics 46 , Academic Press, New York, 1972. · Zbl 0246.57017 [4] V.M. Buchstaber and T.E. Panov: Torus Actions and Their Applications in Topology and Combinatorics, University Lecture Series 24 , Amer. Math. Soc., Providence, RI, 2002. · Zbl 1012.52021 [5] D.A. Cox and S. Katz: Mirror Symmetry and Algebraic Geometry, Mathematical Surveys and Monographs 68 , Amer. Math. Soc., Providence, RI, 1999. · Zbl 0951.14026 [6] W. Chen and Y. Ruan: A new cohomology theory of orbifold , Comm. Math. Phys. 248 (2004), 1–31. · Zbl 1063.53091 [7] M.W. Davis: Smooth $$G$$-manifolds as collections of fiber bundles , Pacific J. Math. 77 (1978), 315–363. · Zbl 0403.57002 [8] M.W. Davis and T. Januszkiewicz: Convex polytopes, Coxeter orbifolds and torus actions , Duke Math. J. 62 (1991), 417–451. · Zbl 0733.52006 [9] R.M. Goresky: Triangulation of stratified objects , Proc. Amer. Math. Soc. 72 (1978), 193–200. · Zbl 0392.57001 [10] A. Hatcher: Algebraic Topology, Cambridge Univ. Press, Cambridge, 2002. · Zbl 1044.55001 [11] A. Hattori and M. Masuda: Theory of multi-fans , Osaka J. Math. 40 (2003), 1–68. · Zbl 1034.57031 [12] M. Masuda and T. Panov: On the cohomology of torus manifolds , Osaka J. Math. 43 (2006), 711–746. · Zbl 1111.57019 [13] J.W. Milnor and J.D. Stasheff: Characteristic Classes, Ann. of Math. Stud. 76 , Princeton Univ. Press, Princeton, N.J., 1974. · Zbl 0298.57008 [14] I. Moerdijk and J. Mrčun: Introduction to Foliations and Lie Groupoids, Cambridge Studies in Advanced Mathematics 91 , Cambridge Univ. Press, Cambridge, 2003. · Zbl 1029.58012 [15] T.E. Panov: Hirzebruch genera of manifolds with torus action , Izv. Ross. Akad. Nauk Ser. Mat. 65 (2001), 123–138, (Russian), translation in Izv. Math. 65 (2001), 543–556. · Zbl 1006.57009 [16] D. Prill: Local classification of quotients of complex manifolds by discontinuous groups , Duke Math. J. 34 (1967), 375–386. · Zbl 0179.12301 [17] P. Scott: The geometries of $$3$$-manifolds , Bull. London Math. Soc. 15 (1983), 401–487. · Zbl 0561.57001 [18] W.P. Thurston: The geometry and topology of three-manifolds , Princeton lecture notes, http://www.msri.org/publications/books/gt3m/.
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