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A local signature for fibered 4-manifolds with a finite group action. (English) Zbl 1297.57051

Let \(\Sigma_g\) be a closed surface of genus \(g\). Let \(G\) be a finite group, and \(p: \Sigma_g \to S^2\) a \(G\)-covering, that is, a finite regular covering whose deck transformation group is isomorphic to \(G\). Let \(E\) and \(B\) be compact oriented manifolds of dimension \(4\) and \(2\), and \(f : E \to B\) a smooth surjective map. A triple \((f, E, B)\) is a fibered \(4\)-manifold of the \(G\)-covering \(p\) if it satisfies (i) \(\partial E = f^{-1}(\partial B)\), (ii) \(f : E \to B\) has finitely many critical values \(\{ b_l \}_{l=1}^{n}\) (the inverse image \(f^{-1} (b_l)\) is called a singular fiber), and the restriction of \(p\) to the complement of singular fibers is a smooth oriented \(\Sigma_g\)-bundle, (iii) the structure group of this \(\Sigma_g\)-bundle is contained in the centralizer \(C(p)\) of the deck transformation group of \(p\) in \(\mathrm{Diff}_+ \Sigma_g\), (iv) the natural \(G\)-action on the complement of singular fibers extends to a smooth action on \(E\).
In this paper, it is shown that, for a \(G\)-covering \(p : \Sigma_g \to S^2\) with at least three branch points, there is a function \(\sigma_{loc}\) from the set of singular fiber germs of fibered 4-manifolds of the \(G\)-covering \(p\) to \(\mathbb{Q}\), such that for any fibered \(4\)-manifolds \((f, E, B)\) of \(G\)-covering, the signature of \(E\) is described as the sum \(\mathrm{Sign}(E) = \sum_{l=1}^{n} \sigma_{loc} (f_l)\), where \(f_l\) is a singular fiber germs of \(f : E \to B\). The function \(\sigma_{loc}\) is explained in Theorem 1.1 (the main theorem of this paper) and proved in Section 3 and 4. For the symmetric mapping class group \(\mathcal{M}_g(p) = \pi_0 C(p)\), the cobounding function of the pull back of the Meyer cocyle is considered to describe the local signature for fibered \(4\)-manifolds of the \(G\)-covering \(p\) when the condition (iv) is ignored and \(\mathcal{M}_g(p)\) satisfies some condition in Section 5. When \(G\) is abelian, the generating set of \(\mathcal{M}_g(p)\) is given in Section 6. When \(G = \mathbb{Z}_d\), for a special type of singular fiber germs, a formula for \(\sigma_{loc}\) is given in Section 7.

MSC:

57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)
14D05 Structure of families (Picard-Lefschetz, monodromy, etc.)
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