Theorem 2.12 [014I]

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Theorem 2.12

If $f:X\rightarrow B$ is as in Theorem 2.1, $B\cong S^{3}$ , and $Crit(f)=\bigcup S_{i}$ as in Definition 2.9, then

(1) each $S_{i}$ is orientable and the canonical orientation is well-defined.

(2) $p_{1}(X)=-2Crit(f)\in H^{4}(X,{\bf Q})$ .

(3) If $X$ has an almost complex structure in which $c_{1}(X)=0$ , then $c_{2}(X)=Crit(f)$ .