Definition 1.1 [0142]

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Definition 1.1

Let $f:X\rightarrow B$ be a continuous, proper mapping of topological manifolds with connected fibres, $\dim X=2n$ , $\dim B=n$ , such that for a dense open set $B_{0}\subseteq B$ , the fibre $X_{b}$ is an $n$ -torus for all $b\in B_{0}$ . We say $f$ is admissible if there exists an open subset $X^{\#}\subseteq X$ with $f^{\#}:X^{\#}\rightarrow B$ the restriction of $f$ to $X^{\#}$ , $Crit(f):=X-X^{\#}$ , satisfying the following properties:

(1) $Sing(X_{b}):=Crit(f)\cap X_{b}$ is a union of a finite number of locally closed submanifolds of dimension at most $n-2$ , and $\Delta:=f(Crit(f))=B-B_{0}$ .

(2) For each $b\in B$ , there exists an open neighbourhood $U\subseteq B$ of $b$ , a rank $n$ vector bundle ${\cal F}$ on $U$ , and an exact sequence

0\rightarrow(R^{n-1}_{c}f^{\#}_{*}{\bf Z})|_{U}\rightarrow{\cal F}\rightarrow f^{\#-1}(U)\rightarrow 0.

Furthermore, the map ${\cal F}\rightarrow f^{\#-1}(U)$ is a local isomorphism.

(3) $R^{n}f_{*}{\bf Z}\cong{\bf Z}$ .