Equivariant Cobordism of Torus Orbifolds

Abstract

Torus orbifolds are topological generalizations of symplectic toric orbifolds. The authours give a construction of smooth orbifolds with torus actions whose boundary is a disjoint union of torus orbifolds using a toric topological method. As a result, they show that any orientable locally standard torus orbifold is equivariantly cobordant to some copies of orbifold complex projective spaces. They also discuss some further equivariant cobordism results including the cases when torus orbifolds are actually torus manifolds.

Keywords

Manifold with corners / Torus action / Torus orbifolds / Equivariant cobordism

References

[1] Adem A, Leida J, Ruan Y. Orbifolds and Stringy Topology, 2007, Cambridge: Cambridge University Press

[2] Cox D A, Katz S. Mirror symmetry and Algebraic Geometry, 1999, Providence, RI: American Mathematical Society

[3] Davis M W. Groups generated by reflections and aspherical manifolds not covered by Euclidean space. Ann. Math., 1983, 117(2): 293-324

[4] Davis M W, Januszkiewicz T. Convex polytopes, Coxeter orbifolds and torus actions. Duke Math. J., 1991, 62(2): 417-451

[5] Druschel K S. Oriented orbifold cobordism. Pacific J. Math., 1994, 164(2): 299-319

[6] Fulton W. Introduction to Toric Varieties, 1993, Princeton, NJ: Princeton University Press

[7] Ganguli S, Poddar M. Almost complex structure, blowdowns and McKay correspondence in quasitoric orbifolds. Osaka J. Math., 2013, 50(4): 977-1005

[8] Grossberg M D, Karshon Y. Equivariant index and the moment map for completely integrable torus actions. Adv. Math., 1998, 133(2): 185-223

[9] Hanke B. Geometric versus homotopy theoretic equivariant bordism. Math. Ann., 2005, 332(3): 677-696

[10] Hattori A, Masuda M. Theory of multi-fans. Osaka J. Math., 2003, 40(1): 1-68

[11] Hirzebruch F. Über eine Klasse von einfachzusammenhängenden komplexen Mannigfaltigkeiten. Math. Ann., 1951, 124: 77-86

[12] Joyce D. On manifolds with corners, Advances in geometric analysis, 2012, Somerville, MA: Int. Press 225-258

[13] Kasprzyk A M. Bounds on fake weighted projective space. Kodai Math. J., 2009, 32(2): 197-208

[14] Katz G. Localization in equivariant bordisms. Math. Z., 1993, 213(4): 617-645

[15] Lerman E, Tolman S. Hamiltonian torus actions on symplectic orbifolds and toric varieties. Trans. Amer. Math. Soc., 1997, 349(10): 4201-4230

[16] Lü Z, Masuda M. Equivariant classification of 2-torus manifolds. Colloq. Math., 2009, 115(2): 171-188

[17] Lü Z, Tan Q. Small covers and the equivariant bordism classification of 2-torus manifolds. Int. Math. Res. Not. IMRN, 2014, 24: 6756-6797

[18] Madsen I. Geometric equivariant bordism and K-theory. Topology, 1986, 25(2): 217-227

[19] Moerdijk I, Mrčun J. Introduction to foliations and Lie groupoids, 2003, Cambridge: Cambridge University Press

[20] Orlik P, Raymond F. Actions of the torus on 4-manifolds, I. Trans. Amer. Math. Soc., 1970, 152: 531-559

[21] Poddar M, Sarkar S. On Quasitoric Orbifolds. Osaka J. Math., 2010, 47(4): 1055-1076

[22] Poddar M, Sarkar S. A class of torus manifolds with nonconvex orbit space. Proc. Amer. Math. Soc., 2015, 143(4): 1797-1811

[23] Pontryagin L S. Gladkie Mnogoobraziya i ih Primeneniya v Teorii Gomotopiĭ, 1955, Moscow: Izdat. Akad. Nauk SSSR

[24] Sarkar S. ${{\mathbb{T}}^2}$-cobordism of quasitoric 4-manifolds. Algebr. Geom. Topol., 2012, 12(4): 2003-2025

[25] Sarkar S. Complex cobordism of quasitoric orbifolds. Topology Appl., 2015, 194: 386-399

[26] Sarkar S, Suh D Y. A new construction of lens spaces. Topology Appl., 2018, 240: 1-20

[27] Satake I. The Gauss-Bonnet theorem for V-manifolds. J. Math. Soc. Japan, 1957, 9: 464-492

[28] Sinha D P. Computations of complex equivariant bordism rings. Amer. J. Math., 2001, 123(4): 577-605

[29] Stong R E. Equivariant bordism and (ℤ₂)^k actions. Duke Math. J., 1970, 37: 779-785

[30] Stong R E. Equivariant bordism and Smith theory, IV. Trans. Amer. Math. Soc., 1976, 215: 313-321

[31] Thom R. Quelques propriétés globales des variétés différentiables. Comment. Math. Helv., 1954, 28: 17-86

[32] tom Dieck T. Bordism of G-manifolds and integrality theorems. Topology, 1970, 9: 345-358