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.
Access provided by Autonomous University of Puebla. Download to read the full chapter text
Chapter PDF
Similar content being viewed by others
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).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Additional information
Dedicated to Professor Lieven Vanhecke
Rights and permissions
Copyright information
© 2005 Birkhäuser Boston
About this chapter
Cite this chapter
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
Download citation
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
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)