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I have the following question: Let $M$ be an even dimensional Riemannian manifold. Under which conditions does there exists a homotopy to some symplectic manifold? is there any chance that such a homotopy exists even if $M$ is not symplectic? how does the homotopy look like? is it differentiable, only continous ... ? is there any chance that $M$ is homotopic to a complex manifold? Is there any reference in this direction ?

greetings mirta

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    $\begingroup$ Do you require a relation between $g$ and $\omega$ (resp. $J$)? If not, your condition that $M$ be Riemannian seems vacuous, since any $M$ is. $\endgroup$
    – Francois Ziegler
    Commented Sep 23, 2013 at 11:35
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    $\begingroup$ Yes, there is a chance that $M$ is homotopy equivalent to a complex manifold. For example, $M$ could be given by forgetting the complex structure on a complex manifold. What question do you really want to ask, and what sort of information do you already have? $\endgroup$
    – S. Carnahan ♦
    Commented Sep 23, 2013 at 11:44
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    $\begingroup$ Homotopy is an equivalence relation among maps. $\endgroup$
    – Fernando Muro
    Commented Sep 23, 2013 at 20:21
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    $\begingroup$ @QiaochuYuan you are implicitly assuming M is closed, because $\mathbb{R}^{2n}$ contradicts you. $\endgroup$
    – Chris Gerig
    Commented Sep 23, 2013 at 21:21
  • $\begingroup$ Ah, so I am. ${}$ $\endgroup$
    – Qiaochu Yuan
    Commented Sep 24, 2013 at 5:27

2 Answers 2

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A result of Szabo implies that there are infinitely many homeomorphic but non-diffeomorphic 4-manifolds which do not admit a symplectic structure (the fact that they are homeomorphic is not explicitly stated, but follows from Freedman's classification). However, they are homeomorphic to a symplectic manifold, in fact a Kahler surface, from Freedman's classification.

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This question is discussed at length in the very nice survey by A. Tralle. (Homotopy properties of closed symplectic manifolds).

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  • $\begingroup$ Igor, isn't Tralle mostly concerned with the distinction between symplectic and Kaehler homotopy-types? For closed simply connected manifolds, the only obstructions to a symplectic form that I know are that the manifold should be c-symplectic (a homotopical property), almost complex (invariant under tangential homotopy-equivalence) and in 4 dimensions should satisfy Taubes's constraints (not homotopical). $\endgroup$
    – Tim Perutz
    Commented Sep 23, 2013 at 16:09
  • $\begingroup$ @TimPerutz he seems to be concerned with both (e.g., he discusses the Thurston/Sullivan conjectures). $\endgroup$
    – Igor Rivin
    Commented Sep 23, 2013 at 18:09
  • $\begingroup$ True, but I don't think he or anyone else has any answers to those questions (except for Seiberg-Witten obstructions in dim 4). In particular, rational homotopy theory hasn't helped us to answer them. $\endgroup$
    – Tim Perutz
    Commented Sep 23, 2013 at 18:27
  • $\begingroup$ @TimPerutz well, the OP asks what is known, and this is what IS known (even if it is not very much :() $\endgroup$
    – Igor Rivin
    Commented Sep 23, 2013 at 19:05

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