Summary
We show that there exist non-formal compact oriented manifolds of dimension n and with first Betti number b 1 = b ≥ 0 if and only if n ≥ 3 and b ≥ 2, or n ≥ (7 − 2b) and 0 ≤ b ≤ 2. Moreover, we present explicit examples for each one of these cases.
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References
Bott, R., Tu, L.W.: Differential forms in algebraic topology. Graduate Texts in Maths, 82. Springer Verlag (1982).
Deligne, P., Griffiths, P., Morgan, J., Sullivan, D.: Real homotopy theory of Kähler manifolds. Invent. Math., 29, 245–274 (1975).
Dranishnikov, A., Rudyak, Y.: Examples of non-formal closed simply connected manifolds of dimensions 7 and more. Preprint math.AT/0306299.
Fernández, M., Gotay, M., Gray, A.: Compact parallelizable four dimensional symplectic and complex manifolds. Proc. Amer. Math. Soc., 103, 1209–1212 (1988).
Fernández, M., Muñoz, V.: Formality of Donaldson submanifolds. Math. Zeit. In press.
Fernández, M., Muñoz, V.: On non-formal simply connected manifolds. Topology and its Appl., 135, 111–117 (2004).
Halperin, S.: Lectures on minimal models. Mém. Soc. Math. France, 230, (1983).
Halperin, S., Gómez-Tato, A., Tanré, D.: Rational homotopy theory for non-simply connected spaces. Trans. Amer. Soc., 352, 1493–1525 (2000).
Lalonde, F., McDuff, D., Polterovich, L.: On the flux conjectures. In: Geometry, topology, and dynamics (Montreal, 1995). CRM Proc. Lecture Notes, 15, 69–85 (1998).
Miller, T.J.: On the formality of (k − 1) connected compact manifolds of dimension less than or equal to (4k − 2). Illinois. J. Math., 23, 253–258 (1979).
Neisendorfer, J., Miller, T.J.: Formal and coformal spaces. Illinois. J. Math., 22, 565–580 (1978).
Oprea, J.: The Samelson space of a fibration. Michigan Math. J., 34, 127–141 (1987).
Tanré, D.: Homotopie rationnelle: Modèles de Chen, Quillen, Sullivan. Lecture Notes in Math., 1025, Springer Verlag (1983).
Tralle, A., Oprea, J.: Symplectic manifolds with no Kähler structure. Lecture Notes in Math., 1661, Springer Verlag (1997).
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Dedicated to Professor Lieven Vanhecke
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Fern↭dez, M., Muñoz, V. (2005). The Geography of Non-Formal Manifolds. In: Kowalski, O., Musso, E., Perrone, D. (eds) Complex, Contact and Symmetric Manifolds. Progress in Mathematics, vol 234. Birkhäuser Boston. https://doi.org/10.1007/0-8176-4424-5_8
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DOI: https://doi.org/10.1007/0-8176-4424-5_8
Publisher Name: Birkhäuser Boston
Print ISBN: 978-0-8176-3850-4
Online ISBN: 978-0-8176-4424-6
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