Abstract
For an orbifold, there are two naturally associated differential graded algebras, one is the de Rham algebra of orbifold differential forms and the other one is the differential graded algebra of piecewise polynomial differential forms of a triangulation of the coarse space. In this paper, we prove that these two differential graded algebras are weakly equivalent; hence, the formality of these two differential graded algebras is consistent, when the triangulation is smooth. We show that global quotient orbifolds and global homogeneous isotropy orbifolds admit smooth triangulations; hence, the two kinds of formality coincide with each other for these orbifolds.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data Availability
Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.
Notes
See for [12, Definition 2.1] for the definition of G-immersion.
References
Adem, A., Leida, J., Ruan, Y.: Orbifolds and stringy topology. Volume 171 of Cambridge Tracts in Mathematics Cambridge University Press, Cambridge (2007)
Babenko, I.K., Taimanov, I.A.: On the existence of informal simply connected symplectic manifolds. Russian Math. Surv. 53(5), 1082–1083 (1998)
Babenko, I.K., Taimanov, I.A.: On nonformal simply-connected symplectic manifolds. Sib. Math. J. 41(2), 204–217 (2000)
Bazzoni, G., Biswas, I., Fernández, M., Muñoz, V., Tralle, A.: Homotopic properties of Kähler orbifolds. In: Chiossi, S.G., Fino, A., Musso, E., Podestá, F., Vezzoni, L. (eds.) Special Metrics and Group Actions in Geometry. Springer INdAM Series, vol. 23, pp. 23–57. Springer, Berlin (2017)
Cavalcanti, G.R., Fernández, M., Muñoz, V.: Symplectic resolutions, Lefschetz property and formality. Adv. Math. 218(2), 576–599 (2008)
Chen, W., Ruan, Y.: A new cohomology theory of orbifold. Commun. Math. Phys. 248(2), 1–31 (2004)
Deligne, P., Griffiths, P., Morgan, J., Sullivan, D.: Real homotopy theory of Kähler manifolds. Invent. Math. 29(3), 245–274 (1975)
Du, C.-Y., Shen, L., Zhao, X.: Spark complexes on good effective orbifold atlases categorically. Theory Appl. Categ. 33(26), 784–812 (2018)
Félix, Y., Halperin, S., Thomas, J.-C.: Rational Homotopy Theory. Graduate Texts in Mathematics, vol. 205. Springer, New York (2001)
Fernández, M., Muñoz, V.: An \(8\)-dimensional non-formal simply connected symplectic manifold. Ann. Math. 2(167), 1045–1054 (2008)
Griffiths, P., Morgan, J.: Rational Homotopy Theory and Differential Forms. Progress in Mathematics, vol. 16. Springer, New York (2013)
Illman, S.: Smooth equivariant triangulations of \(G\)-manifolds for \(G\) a finite group. Math. Ann. 233, 199–220 (1978)
Lupton, G., Oprea, J.: Symplectic manifolds and formality. J. Pure Appl. Algebra 91(1–3), 193–207 (1994)
McDuff, D.: Examples of simply-connected symplectic non-Kähler manifolds. J. Differential Geom. 20, 267–277 (1984)
Moerdijk, I., Pronk, D.: Orbifolds, sheaves and groupoids. K-Theory 12, 3–21 (1997)
Moerdijk, I., Pronk, D.: Simplicial cohomology of orbifolds. Indag. Math. (N.S.) 10(2), 269–293 (1999)
Munkres, J.: Elementary Differential Topology, vol. 53. Princeton University Press, Princeton (1963)
Muñoz, V., Rojo, J.A.: Symplectic resolution of orbifolds with homogeneous isotropy. Geom. Dedicata. 204(1), 339–363 (2020)
Neisendorfer, J., Miller, T.: Formal and coformal spaces. Illinois J. Math. 22(4), 565–580 (1978)
Satake, I.: On a generalization of the notion of manifold. Proc. Natl. Acad. Sci. U.S.A. 42(6), 359–363 (1956)
Sullivan, D.: Infinitesimal computations in topology. Publ. Math. Inst. Hautes Études Sci. 47, 269–331 (1977)
Acknowledgements
The authors thank the anonymous referee for his/her valuable advices and professor Bohui Chen for valuable discussion on triangulations of orbifolds.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This work was supported by the National Natural Science Foundation of China (Grant No. 12071322) and by Sichuan Science and Technology Program (Grant No. 2022JDTD0019)
Rights and permissions
Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Du, CY., He, K. & Xue, H. On triangulations of orbifolds and formality. Ann Glob Anal Geom 62, 829–845 (2022). https://doi.org/10.1007/s10455-022-09874-w
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10455-022-09874-w