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The homotopy classification of four-dimensional toric orbifolds

Part of: Differential topology Homotopy theory

Published online by Cambridge University Press:  25 May 2021

Xin Fu ,
Tseleung So  and
Jongbaek Song Open the ORCID record for Jongbaek Song [Opens in a new window]
Xin Fu
Affiliation:
Department of Mathematics, Ajou University, Suwon 16499, Republic of Korea ([email protected])
Tseleung So
Affiliation:
Department of Mathematics and Statistics, University of Regina, Regina, SK S4S 0A2, Canada ([email protected])
Jongbaek Song
Affiliation:
School of Mathematics, KIAS, Seoul 02455, Republic of Korea ([email protected])
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Abstract

Let X be a 4-dimensional toric orbifold. If $H^{3}(X)$ has a non-trivial odd primary torsion, then we show that X is homotopy equivalent to the wedge of a Moore space and a CW-complex. As a corollary, given two 4-dimensional toric orbifolds having no 2-torsion in the cohomology, we prove that they have the same homotopy type if and only their integral cohomology rings are isomorphic.

Keywords

Cohomological rigiditytoric orbifold

MSC classification

Primary: 57R18: Topology and geometry of orbifolds 55P15: Classification of homotopy type
Secondary: 55P60: Localization and completion
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 152 , Issue 3 , June 2022 , pp. 626 - 648
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Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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References

Bukhshtaber, V. M., Erokhovets, N. Y., Masuda, M., Panov, T. E. and Pak, S.. Cohomological rigidity of manifolds defined by 3-dimensional polytopes. Uspekhi Mat. Nauk. 72 (2017), 3–66.Google Scholar
Bahri, A., Franz, M., Notbohm, D. and Ray, N.. The classification of weighted projective spaces. Fund. Math. 220 (2013), 217–226.CrossRefGoogle Scholar
Bahri, A., Notbohm, D., Sarkar, S. and Song, J.. On integral cohomology of certain orbifolds. Int. Math. Res. Not. IMRN. 6 (2021), 4140–4168.CrossRefGoogle Scholar
Bahri, A., Sarkar, S. and Song, J.. On the integral cohomology ring of toric orbifolds and singular toric varieties. Algebr. Geom. Topol. 17 (2017), 3779–3810.CrossRefGoogle Scholar
Choi, S.. Classification of Bott manifolds up to dimension 8. Proc. Edinb. Math. Soc. (2) 58 (2015), 653–659.10.1017/S0013091514000121CrossRefGoogle Scholar
Choi, S., Masuda, M. and Suh, D. Y.. Topological classification of generalized Bott towers. Trans. Am. Math. Soc. 362 (2010), 1097–1112.CrossRefGoogle Scholar
Cox, D. A., Little, J. B. and Schenck, H. K.. Toric varieties, volume 124 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2011.CrossRefGoogle Scholar
Darby, A., Kuroki, S. and Song, J.. The equivariant cohomology of torus orbifolds. To appear in. Can. J. Math. (2020), 1–30. doi:10.4153/S0008414X20000760.Google Scholar
Davis, M. W. and Januszkiewicz, T.. Convex polytopes, Coxeter orbifolds and torus actions. Duke Math. J. 62 (1991), 417–451.CrossRefGoogle Scholar
Fischli, S.. On toric varieties. Ph.D. thesis, Universität Bern, 1992.Google Scholar
Freedman, M. H.. The topology of four-dimensional manifolds. J. Diff. Geom. 17 (1982), 357–453.Google Scholar
Hatcher, A.. Algebraic Topology (Cambridge: Cambridge University Press, 2002).Google Scholar
Jordan, A.. Homology and cohomology of toric varieties. Ph.D. thesis, University of Konstanz, 1998.Google Scholar
Kuwata, H., Masuda, M. and Zeng, H.. Torsion in the cohomology of torus orbifolds. Chin. Ann. Math. Ser. B 38 (2017), 1247–1268.10.1007/s11401-017-1034-4CrossRefGoogle Scholar
Masuda, M. and Suh, D. Y.. Classification problems of toric manifolds via topology. In Toric topology, volume 460 of Contemp. Math., pp. 273–286. Am. Math. Soc., Providence, RI, 2008.10.1090/conm/460/09024CrossRefGoogle Scholar
Orlik, P. and Raymond, F.. Actions of the torus on 4-manifolds. I. Trans. Am. Math. Soc. 152 (1970), 531–559.Google Scholar
Porter, G. J.. The homotopy groups of wedges of suspensions. Am. J. Math. 88 (1966), 655–663.CrossRefGoogle Scholar
Poddar, M. and Sarkar, S.. On quasitoric orbifolds. Osaka J. Math. 47 (2010), 1055–1076.Google Scholar
Sarkar, S. S. and Suh, D. Y.. A new construction of lens spaces. Topology Appl. 240 (2018), 1–20.CrossRefGoogle Scholar
Selick, P.. Introduction to homotopy theory, volume 9 of Fields Institute Monographs. American Mathematical Society, Providence, RI, 1997.Google Scholar
So, T. and Theriault, S.. The suspension of a 4-manifold and its applications. arXiv:1909.11129, 2019.Google Scholar