##
**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.

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.) |

### Citations:

Zbl 0592.53025### References:

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