On deformations of compact foliations
HTML articles powered by AMS MathViewer
- by Matias del Hoyo and Rui Loja Fernandes PDF
- Proc. Amer. Math. Soc. 147 (2019), 4555-4561 Request permission
Abstract:
We combine classic stability results for foliations with recent results on deformations of Lie groupoids and Lie algebroids to provide a cohomological characterization for rigidity of compact Hausdorff foliations on compact manifolds.References
- Camilo Arias Abad and Marius Crainic, Representations up to homotopy of Lie algebroids, J. Reine Angew. Math. 663 (2012), 91–126. MR 2889707, DOI 10.1515/CRELLE.2011.095
- Marius Crainic, Differentiable and algebroid cohomology, van Est isomorphisms, and characteristic classes, Comment. Math. Helv. 78 (2003), no. 4, 681–721. MR 2016690, DOI 10.1007/s00014-001-0766-9
- M. Crainic, R. L. Fernandes, and D. Martinez Torres, Regular Poisson manifolds of compact types (PMCT 2), preprint, arXiv:1603.00064, 2016.
- Marius Crainic and Ieke Moerdijk, Deformations of Lie brackets: cohomological aspects, J. Eur. Math. Soc. (JEMS) 10 (2008), no. 4, 1037–1059. MR 2443928, DOI 10.4171/JEMS/139
- M. Crainic, J. N. Mestre, and I. Struchiner, Deformations of Lie groupoids, preprint, arXiv: 1510.02530, 2015.
- Matias del Hoyo and Rui Loja Fernandes, Riemannian metrics on Lie groupoids, J. Reine Angew. Math. 735 (2018), 143–173. MR 3757473, DOI 10.1515/crelle-2015-0018
- M. del Hoyo and R. L. Fernandes, Riemannian metrics on differentiable stacks, preprint, arXiv:1601.05616, Mathematische Zeitschrift (to appear).
- D. B. A. Epstein, Foliations with all leaves compact, Ann. Inst. Fourier (Grenoble) 26 (1976), no. 1, viii, 265–282 (English, with French summary). MR 420652
- D. B. A. Epstein and H. Rosenberg, Stability of compact foliations, Geometry and topology (Proc. III Latin Amer. School of Math., Inst. Mat. Pura Aplicada CNPq, Rio de Janeiro, 1976) Lecture Notes in Math., Vol. 597, Springer, Berlin, 1977, pp. 151–160. MR 0501007
- R. Hamilton, Deformation theory of foliations, preprint, Cornell University, 1978.
- James L. Heitsch, A cohomology for foliated manifolds, Comment. Math. Helv. 50 (1975), 197–218. MR 372877, DOI 10.1007/BF02565746
- Rémi Langevin and Harold Rosenberg, Integrable perturbations of fibrations and a theorem of Seifert, Differential topology, foliations and Gelfand-Fuks cohomology (Proc. Sympos., Pontifícia Univ. Católica, Rio de Janeiro, 1976) Lecture Notes in Mathematics, Vol. 652, Springer, Berlin, 1978, pp. 122–127. MR 505655
- I. Moerdijk and J. Mrčun, Introduction to foliations and Lie groupoids, Cambridge Studies in Advanced Mathematics, vol. 91, Cambridge University Press, Cambridge, 2003. MR 2012261, DOI 10.1017/CBO9780511615450
- William P. Thurston, A generalization of the Reeb stability theorem, Topology 13 (1974), 347–352. MR 356087, DOI 10.1016/0040-9383(74)90025-1
Additional Information
- Matias del Hoyo
- Affiliation: Departamento de Geometria - IME, Universidade Federal Fluminense, Rua Professor Marcos Waldemar de Freitas Reis, 24210-201, Niterói, Brazil
- Email: mldelhoyo@id.uff.br
- Rui Loja Fernandes
- Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 W. Green Street, Urbana, Illinois 61801
- MR Author ID: 341522
- Email: ruiloja@illinois.edu
- Received by editor(s): July 27, 2018
- Received by editor(s) in revised form: January 21, 2019, January 27, 2019, January 28, 2019, and January 30, 2019
- Published electronically: July 1, 2019
- Additional Notes: The first author was supported by National Council for Scientific and Technological Development - CNPq grant 303034/2017-3.
The second author was partially supported by NSF grants DMS 1405671, DMS 1710884 and a Simons Fellowship.
Both authors acknowledge the support of the Ciências Sem Fronteiras grant 401817/2013-0. - Communicated by: Jiaping Wang
- © Copyright 2019 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 147 (2019), 4555-4561
- MSC (2010): Primary 22A22, 53C12
- DOI: https://doi.org/10.1090/proc/14567
- MathSciNet review: 4002563