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Examples of symplectic structures. (English) Zbl 0625.53040
The author constructs a family $$\{\tilde w_ k$$; $$k\in Z$$, $$k\geq 0\}$$ of symplectic forms on a compact manifold $$\tilde Y$$ which have the same homotopy theoretic invariants, but which are not diffeomorphic. The construction, inspired by M. Gromov [Invent. Math. 82, 307-347 (1985; Zbl 0592.53025)] can be summarized as follows.
Let $$Y=S^ 2\times T^ 2\times S^ 2\times S^ 2$$, $$Z=S^ 2\times S^ 2$$ and $$i: Z\hookrightarrow Y$$ the symplectic embedding $$i(z)=(z,w_ 0,u_ 0)$$. Now, let $$\tilde Y=S^ 2\times T^ 2\times (S^ 2\times S^ 2\#\overline{CP}^ 2)$$ be the blow up of Y along i(Z). Then it is possible to construct a family $$\{\tilde w_ k$$; $$k\in Z\}$$ of symplectic forms on $$\tilde Y$$ such that
(1) there is a family $$\{\tilde w_ t$$; $$t\in R\}$$ of symplectic forms on $$\tilde Y$$ which extends the family $$\{\tilde w_ k$$; $$k\in Z\};$$
(2) the forms $$\tilde w_ k$$ and $$\tilde w_{-k}$$ are diffeomorphic for each $$k\in Z;$$
(3) no two of the forms $$\tilde w_ k$$, $$k\geq 0$$ are diffeomorphic.
To prove (3), the author shows that there exists a particular family of pseudo-holomorphic curves which twists around another, the twisting being measured by a generalized Hopf invariant.
Reviewer: M.de Leon

##### MSC:
 53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.) 57R15 Specialized structures on manifolds (spin manifolds, framed manifolds, etc.)
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##### References:
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