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On symplectic cobordisms. (English. Russian original) Zbl 0216.20201

Math. USSR, Sb. 12(1970), 77-89 (1971); translation from Mat. Sb., n. Ser. 83(125), 77-89 (1970).
Summary: In the article, the method of spherical reconstructions of smooth manifolds is applied to the computation of some groups of symplectic cobordisms. Namely, it is proved that \(\Omega^5_{Sp} = \mathbb Z_2\), \(\Omega^6_{Sp} = \mathbb Z_2\) and \(\Omega^7_{Sp} = 0\). The indicated values of the groups of cobordisms for dimensions 5 and 6 are known and follow from arguments of the Adams spectral sequence for \(Sp\)-cobordisms. The new result is the fact that the seventh group of cobordisms equals 0. This is the fundamental result of the article. The theorem concerning the reconstruction of manifolds with a quasisymplectic structure in the normal bundle, which is proved in the article, and the theorem on integer values of Atiyah-Hirzebruch constitute the basis for the proof.

MSC:

57N70 Cobordism and concordance in topological manifolds
57R17 Symplectic and contact topology in high or arbitrary dimension
57R90 Other types of cobordism
22Exx Lie groups
55R10 Fiber bundles in algebraic topology
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