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Homotopy properties of symplectic blow-ups. (English) Zbl 1066.57029

Albuquerque, H. (ed.) et al., Proceedings of the XII fall workshop on geometry and physics, Coimbra, Portugal, September 8–10, 2003. Madrid: Real Sociedad Matemática Española (ISBN 84-933610-3-8/pbk). Publicaciones de la Real Sociedad Matemática Española 7, 95-109 (2004).
In a previous paper [Math. Z. 250, 149–175 (2005; Zbl 1071.57024)] the authors have introduced the concept of \(s\)-formality as a weaker version of formality (in the sense of P. Deligne, P. Griffiths, J. Morgan and D. Sullivan [Invent. Math. 29, 245–274 (1975; Zbl 0312.55011)]). They have proved that a connected and orientable compact differentiable manifold of dimension \(2n\), or \((2n- 1)\) is formal if and only if it is \((n- 1)\)-formal and that any simply connected compact manifold \(M\) is \(2\)-formal.
In this paper the authors analyze the relationship between the \(s\)-formality of any compact symplectic manifold \(M\) and the \(s\)-formality of the symplectic blow-up \(\widetilde{\mathbb C P^m}\) of the complex projective space \(\mathbb C P^m\) along a symplectic embedding \(M\hookrightarrow \mathbb C P^m\). More precisely, the authors show that if \(M\) is \(s\)-formal then \(\widetilde{\mathbb C P^m}\) is \((s+ 2)\)-formal. Also if \(M\) is formal then \(\widetilde{\mathbb C P^m}\) is formal. This study permits to the authors to give examples of simply connected compact symplectic manifolds which are \(s\)-formal but not \((s+ 1)\)-formal for arbitrarily large values of \(s\). The authors also discuss specific examples for small values of \(s\).
For the entire collection see [Zbl 1055.53001].
Reviewer: Ioan Pop (Iaşi)

MSC:

57R17 Symplectic and contact topology in high or arbitrary dimension
55P62 Rational homotopy theory
53D35 Global theory of symplectic and contact manifolds
55S30 Massey products
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