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See my previous question here.

Let $M$ be a smooth closed simply-connected $4$-manifold with $w_1 = w_2 = 0$. Can $TM$ be trivialized in the complement of a point?

This was answered in the affirmative.

My question now is, is it possible to conclude in some way from Smale's immersion theorem that $M - p$ admits a symplectic structure, where $p$ is a point? I see how to do it with Gromov's $h$-principal, but not with Smale's immersion theorem.

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    $\begingroup$ This is actually trivial (and you do not need simple connectivity): Take an immersion $M\to R^4$ and pull-back the symplectic structure from $R^4$ to $M$. $\endgroup$
    – Moishe Kohan
    Commented Feb 26, 2016 at 22:17

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By an application of Gromov's $h$-principle, an open manifold $W$ admits a symplectic structure precisely if $TW$ admits an almost complex structure. In the case you are considering it does, as $TW$ is trivial.

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  • $\begingroup$ Thanks, but is there an easier way to see this that does not invoke something as powerful as Gromov's $h$-principle? $\endgroup$
    – user74565
    Commented Feb 26, 2016 at 20:10
  • $\begingroup$ @Antonio: The h-principle for open $\text{Diff}(V)$-invariant relations is not so terrifying. There's a short proof in Eliashberg and Mishachev's book. $\endgroup$
    – mme
    Commented Feb 26, 2016 at 20:13

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