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The following is taken from Audin, Damian: Morse Theory and Floer Homology:

Assumption 6.2.2 from the book

My questions about this:

Question 1:

I understand "exists a symplectic trivialization" as: There exists a symplectic vector space $F$ and an isomorphism of symplectic fiber bundles $f:\omega ^* TW \rightarrow S^2 \times F$. Is that what is meant?

Question 2:

The vector bundle (with complex structure given by a symplectic structure and some metric on $W$) $\omega ^*TW$ being trivial is equivalent to $c_k(\omega^*TW)=0$ for $k>0$. Why the shorthand $\langle c_1(TW), \pi_2(W) \rangle =0$ then? Why is nothing said about the other Chern classes?

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1 Answer 1

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Question 1: Yes.

Question 2: Note that $c_{i}(\omega^{*}TW) \in H^{2i}(S^{2},\mathbb{Z})$. For $i>1$ we have that $H^{2i}(S^{2},\mathbb{Z}) = 0$ (because $S^{2}$ is a $2$-dimensional manifold), hence the chern class $c_{i}$ vanishes for $i>1$. This is why there is no reference to the other Chern classes.

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