Home Mathematics The Stiefel–Whitney classes of moment-angle manifolds are trivial
Article
Licensed
Unlicensed Requires Authentication

The Stiefel–Whitney classes of moment-angle manifolds are trivial

  • Sho Hasui EMAIL logo , Daisuke Kishimoto and Akatsuki Kizu
Published/Copyright: October 26, 2022

Abstract

We prove that the Stiefel–Whitney classes of a moment-angle manifold, not necessarily smooth, are trivial. We also consider Stiefel–Whitney classes of the partial quotient of a moment-angle manifold.

MSC 2010: 57N65; 55N91

Communicated by Jan Bruinier


Award Identifier / Grant number: JP18K13414

Award Identifier / Grant number: JP17K05248

Award Identifier / Grant number: JP19K03473

Acknowledgements

The authors are grateful to the referee for useful advice and comments. The first author was supported by JSPS KAKENHI Grant no. JP18K13414, and the second author was supported by JSPS KAKENHI Grant nos. JP17K05248 and JP19K03473.

References

[1] Y. Barreto, S. López de Medrano and A. Verjovsky, Some open book and contact structures on moment-angle manifolds, Bol. Soc. Mat. Mex. (3) 23 (2017), no. 1, 423–437. 10.1007/s40590-016-0113-ySearch in Google Scholar

[2] V. M. Buchstaber and T. E. Panov, Torus Actions and Their Applications in Topology and Combinatorics, Univ. Lecture Ser. 24, American Mathematical Society, Providence, 2002. 10.1090/ulect/024Search in Google Scholar

[3] L. Cai, On products in a real moment-angle manifold, J. Math. Soc. Japan 69 (2017), no. 2, 503–528. 10.2969/jmsj/06920503Search in Google Scholar

[4] M. Chaperon and S. López De Medrano, Birth of attracting compact invariant submanifolds diffeomorphic to moment-angle manifolds in generic families of dynamics, C. R. Math. Acad. Sci. Paris 346 (2008), no. 19–20, 1099–1102. 10.1016/j.crma.2008.09.017Search in Google Scholar

[5] D. A. Cox, The homogeneous coordinate ring of a toric variety, J. Algebraic Geom. 4 (1995), no. 1, 17–50. 10.1090/S1056-3911-2013-00651-7Search in Google Scholar

[6] M. W. Davis and T. Januszkiewicz, Convex polytopes, Coxeter orbifolds and torus actions, Duke Math. J. 62 (1991), no. 2, 417–451. 10.1215/S0012-7094-91-06217-4Search in Google Scholar

[7] E. Fadell, Generalized normal bundles for locally-flat imbeddings, Trans. Amer. Math. Soc. 114 (1965), 488–513. 10.1090/S0002-9947-1965-0179795-4Search in Google Scholar

[8] M. Franz, The cohomology rings of smooth toric varieties and quotients of moment-angle complexes, Geom. Topol. 25 (2021), no. 4, 2109–2144. 10.2140/gt.2021.25.2109Search in Google Scholar

[9] S. Gitler and S. López de Medrano, Intersections of quadrics, moment-angle manifolds and connected sums, Geom. Topol. 17 (2013), no. 3, 1497–1534. 10.2140/gt.2013.17.1497Search in Google Scholar

[10] S. Hasui, On the classification of quasitoric manifolds over dual cyclic polytopes, Algebr. Geom. Topol. 15 (2015), no. 3, 1387–1437. 10.2140/agt.2015.15.1387Search in Google Scholar

[11] I. Y. Limonchenko, Massey products in the cohomology of moment-angle manifolds of 2-truncated cubes (in Russian), Uspekhi Mat. Nauk 71 (2016), no. 2(428), 207–208; translation in Russian Math. Surveys 71 (2016), no. 2, 376-378. 10.1070/RM9712Search in Google Scholar

[12] M. Mimura and H. Toda, Topology of Lie Groups. I, II, Transl. Math. Monogr. 91, American Mathematical Society, Providence, 1991. Search in Google Scholar

[13] A. E. Mironov and T. E. Panov, Intersections of quadrics, moment-angle manifolds, and Hamiltonian-minimal Lagrangian embeddings (in Russian), Funktsional. Anal. i Prilozhen. 47 (2013), no. 1, 47–61; translation in Funct. Anal. Appl. 47 (2013), no. 1, 38-49. 10.1007/s10688-013-0005-0Search in Google Scholar

[14] R. S. Palais, The classification of G-spaces, Mem. Amer. Math. Soc. 36 (1960), 1–72. 10.1090/memo/0036Search in Google Scholar

[15] T. Panov, Y. Ustinovskiy and M. Verbitsky, Complex geometry of moment-angle manifolds, Math. Z. 284 (2016), no. 1–2, 309–333. 10.1007/s00209-016-1658-1Search in Google Scholar

[16] T. E. Panov, On the cohomology of quotient spaces of moment-angle complexes (in Russian), Uspekhi Mat. Nauk 70 (2015), no. 4(424), 209–210; translation in Russian Math. Surveys 70 (2015), no. 4, 779-781. Search in Google Scholar

[17] P. Sankaran, Determination of Grassmann manifolds which are boundaries, Canad. Math. Bull. 34 (1991), no. 1, 119–122. 10.4153/CMB-1991-019-8Search in Google Scholar

[18] S. Theriault, Moment-angle manifolds and Panov’s problem, Int. Math. Res. Not. IMRN 2015 (2015), no. 20, 10154–10175. 10.1093/imrn/rnu281Search in Google Scholar

[19] G. M. Ziegler, Lectures on Polytopes, Grad. Texts in Math. 152, Springer, New York, 1995. 10.1007/978-1-4613-8431-1Search in Google Scholar

Received: 2021-10-17
Revised: 2022-09-06
Published Online: 2022-10-26
Published in Print: 2022-11-01

© 2022 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 20.12.2025 from https://www.degruyterbrill.com/document/doi/10.1515/forum-2021-0267/html
Scroll to top button