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Non-formal co-symplectic manifolds

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2015
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American Mathematical Society
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We study the formality of the mapping torus of an orientation-preserving diffeomorphism of a manifold. In particular, we give conditions under which a mapping torus has a non-zero Massey product. As an application we prove that there are non-formal compact co-symplectic manifolds of dimension m and with first Betti number b if and only if m = 3 and b >= 2, or m >= 5 and b >= 1. Explicit examples for each one of these cases are given.
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[1] I. K. Babenko and I. A. Taĭmanov, On nonformal simply connected symplectic manifolds, Sibirsk. Mat. Zh. 41 (2000), no. 2, 253–269, i (Russian, with Russian summary); English transl., Siberian Math. J. 41 (2000), no. 2, 204–217. MR 1762178 (2001g:57051), [2] Giovanni Bazzoni and Vicente Muñoz, Classification of minimal algebras over any field up to dimension 6, Trans. Amer. Math. Soc. 364 (2012), no. 2, 1007–1028. MR 2846361 (2012i:55010), [3] D. E. Blair, The theory of quasi-Sasakian structures, J. Differential Geometry 1 (1967), 331–345. MR 0226538 (37 #2127) [4] David E. Blair, Contact manifolds in Riemannian geometry, Lecture Notes in Mathematics, Vol. 509, Springer-Verlag, Berlin-New York, 1976. MR 0467588 (57 #7444) [5] David E. Blair, Riemannian geometry of contact and symplectic manifolds, Progress in Mathematics, vol. 203, Birkhäuser Boston, Inc., Boston, MA, 2002. MR 1874240 (2002m:53120) [6] Raoul Bott and Loring W. Tu, Differential forms in algebraic topology, Graduate Texts in Mathematics, vol. 82, Springer-Verlag, New York-Berlin, 1982. MR 658304 (83i:57016) [7] D. Chinea, M. de León, and J. C. Marrero, Topology of cosymplectic manifolds, J. Math. Pures Appl. (9) 72 (1993), no. 6, 567–591. MR 1249410 (95c:53036) [8] Luis A. Cordero, M. Fernández, and A. Gray, Symplectic manifolds with no Kähler structure, Topology 25 (1986), no. 3, 375–380. MR 842431 (87j:53051), [9] Pierre Deligne, Phillip Griffiths, John Morgan, and Dennis Sullivan, Real homotopy theory of Kähler manifolds, Invent. Math. 29 (1975), no. 3, 245–274. MR 0382702 (52 #3584) [10] Yves Félix, Stephen Halperin, and Jean-Claude Thomas, Rational homotopy theory, Graduate Texts in Mathematics, vol. 205, Springer-Verlag, New York, 2001. MR 1802847 (2002d:55014) [11] Yves Félix, John Oprea, and Daniel Tanré, Algebraic models in geometry, Oxford Graduate Texts in Mathematics, vol. 17, Oxford University Press, Oxford, 2008. MR 2403898 (2009a:55006) [12] Marisa Fernández, Alfred Gray, and John W. Morgan, Compact symplectic manifolds with free circle actions, and Massey products, Michigan Math. J. 38 (1991), no. 2, 271–283. MR 1098863 (92e:57049), [13] Marisa Fernández and Vicente Muñoz, Formality of Donaldson submanifolds, Math. Z. 250 (2005), no. 1, 149–175. MR 2136647 (2006a:53098), [14] Marisa Fernández and Vicente Muñoz, Non-formal compact manifolds with small Betti numbers, Contemporary geometry and related topics, Univ. Belgrade Fac. Math., Belgrade, 2006, pp. 231–246. MR 2963633 [15] Marisa Fernández and Vicente Muñoz, An 8-dimensional nonformal, simply connected, symplectic manifold, Ann. of Math. (2) 167 (2008), no. 3, 1045–1054. MR 2415392 (2009j:53114) [16] Anna Fino and Luigi Vezzoni, Some results on cosymplectic manifolds, Geom. Dedicata 151 (2011), 41–58. MR 2780737 (2012c:53042) [17] Samuel I. Goldberg and Kentaro Yano, Integrability of almost cosymplectic structures, Pacific J. Math. 31 (1969), 373–382. MR 0251678 (40 #4905) [18] Akio Hattori, Spectral sequence in the de Rham cohomology of fibre bundles, J. Fac. Sci. Univ. Tokyo Sect. I 8 (1960), 289–331 (1960). MR 0124918 (23 #A2226) [19] Hongjun Li, Topology of co-symplectic/co-Kähler manifolds, Asian J. Math. 12 (2008), no. 4, 527–543. MR 2481690 (2010c:53126) [20] Paulette Libermann, Sur les automorphismes infinitésimaux des structures symplectiques et des structures de contact, Colloque Géom. Diff. Globale (Bruxelles, 1958) Centre Belge Rech. Math., Louvain, 1959, pp. 37–59 (French). MR 0119153 (22 #9919) [21] L. Magnin, Sur les algèbres de Lie nilpotentes de dimension ≤7, J. Geom. Phys. 3 (1986), no. 1, 119–144 (French, with English summary). MR 855573 (87k:17012) [22] G. D. Mostow, Errata, “Factor spaces of solvable groups.”, Ann. of Math. (2) 66 (1957), 590. MR 0090001 (19,752d) [23] Joseph Neisendorfer and Timothy Miller, Formal and coformal spaces, Illinois J. Math. 22 (1978), no. 4, 565–580. MR 0500938 (58 #18429) [24] Aleksy Tralle and John Oprea, Symplectic manifolds with no Kähler structure, Lecture Notes in Mathematics, vol. 1661, Springer-Verlag, Berlin, 1997. MR 1465676 (98k:53038) [25] D. Tischler, On fibering certain foliated manifolds over
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