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Let $(M,\omega)$ and $(M^{\prime},\omega^{\prime})$ are two symplectic manifolds. Then we may define a natural homotopy equivalence as follows.

We say that the smooth map $f:M\longrightarrow M^{\prime}$ is $\textbf{symplectic}$, if $f^{*}\omega^{\prime}=\omega$.

$\textbf{Definition}$: We say that the symplectic map $f:(M,\omega)\longrightarrow (M^{\prime},\omega^{\prime})$ is a symplectic homotopy equivalence, if there exists a symplectic map $g:(M^{\prime},\omega^{\prime})\longrightarrow (M,\omega)$ such that $g\circ f\simeq id_{M}$ and $f\circ g\simeq id_{M^{\prime}}$.

I want to search some papers that consider such symplectic homotopy equivalence, can somebody help me?

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    $\begingroup$ Strange definition. At least, do you want them homotopic in the class of symplectic maps? $\endgroup$ Nov 9, 2014 at 13:51

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If a map $f: M\to M'$ is symplectic, then it's a local diffeomorphism, since the pull back of the volume form $\omega'\wedge \dots \wedge \omega'$ is a volume form (namely, $\omega^{\wedge n}$). Hence it's a covering map, and therefore your definition of symplectic homotopy equivalence is equivalent to the definition of symplectomorphism (at the very least, for compact manifolds).

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  • $\begingroup$ Thanks very much. I wonder why nobody consider this case, it seems a natural concept in symplectic category. $\endgroup$ Nov 10, 2014 at 15:04

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