Definition 2.9 [014H]

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Definition 2.9

(1) If $g:X^{\prime}\rightarrow D$ is a $T^{2}$ -fibration over a disk as constructed in Example 2.4, (3), there is an immersion $i:S^{2}\rightarrow X^{\prime}$ onto the singular fibre $X_{0}^{\prime}$ . This immersion fails to be an embedding precisely at the singular point of $X_{0}^{\prime}$ , where two sheets of the immersed $S^{2}$ cross. This local structure can be seen in the map $f:{\bf C}^{2}\rightarrow{\bf R}^{2}$ of Example 2.3, or equivalently, in the map ${\bf C}^{2}\rightarrow{\bf C}$ given by $(z_{1},z_{2})\mapsto z_{1}z_{2}$ . Given an orientation on $S^{2}$ and on $X^{\prime}$ , we call the orientation on $X^{\prime}$ for which these two sheets intersect positively the positive orientation on $X^{\prime}$ . Note this is independent of the choice of orientation on $S^{2}$ .

(2) Given $f:X\rightarrow B$ a $T^{3}$ -fibration produced by Theorem 2.1, and an orientation on $X$ , then $Crit(f)$ is a union of connected two-manifolds $\bigcup S_{i}$ meeting at most at points. We can orient each $S_{i}$ as follows. For a point $b\in\Delta_{g}\cap f(S_{i})$ , there is a neighbourhood $U=D\times(0,1)$ of $b$ such that $f^{-1}(U)=X^{\prime}\times(0,1)\times S^{1}$ , so that $f$ is induced by a map $g:X^{\prime}\rightarrow D$ , as in Definition 1.2. Take the positive orientation on $X^{\prime}$ over $U$ . Then $S_{i}\cap f^{-1}(U)$ is the surface $S=Crit(g)\times(0,1)\times S^{1}$ meeting $X^{\prime}\times\{1/2\}\times\{p\}$ transversally. Orient $S$ so that it meets this latter surface positively. If each $S_{i}$ is orientable, this gives an orientation on $S_{i}$ for each $i$ , and hence makes $Crit(f)$ into an oriented two-cycle. We call this orientation on $Crit(f)$ the canonical orientation. We shall see in Theorem 2.12 that this orientation does not depend on the choice of $b\in\Delta_{g}\cap f(S_{i})$ , and that $S_{i}$ is orientable.