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Odd sphere bundles, symplectic manifolds, and their intersection theory. (English) Zbl 1406.53085

Motivated by mirror symmetry between complex and symplectic manifolds, L.-S. Tseng and S.-T. Yau [J. Differ. Geom. 91, No. 3, 383–416 (2012; Zbl 1275.53079); J. Differ. Geom. 91, No. 3, 417–443 (2012; Zbl 1275.53080)] introduced, for each \(p=0,1,\dots,\dim(M)\), the \(p\)-filtered cohomology \(F^pH^*(M)\) for every symplectic manifold \((M,\omega)\). The \(p\)-filtered cohomology is expected to be important in the study of deformation of symplectic manifolds. Later on, C.-J. Tsai et al. [J. Differ. Geom. 103, No. 1, 83–143 (2016; Zbl 1353.53085)] showed that each \(F^pH^*(M)\) is the cohomology of a not necessarily formal \(A_{\infty}\) algebra \(\mathcal{F}_p\). The paper under review provides a topological origin to this \(A_{\infty}\) algebra. In particular, when \(\omega\) is integral, the authors relate \(\mathcal{F}_p\) to the de Rham differential graded algebra of the associated odd sphere bundle of \(M\). Some applications are given, including the Calabi-Yau property of \(\mathcal{F}_p\).

MSC:

53D35 Global theory of symplectic and contact manifolds
57R17 Symplectic and contact topology in high or arbitrary dimension
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