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Signatures of singular branched covers. (English) Zbl 0793.57013
We consider branched cyclic \(d\)-fold covers of a closed smooth oriented manifold \(M\) along a collection of codimension-2 smooth oriented submanifolds \(\{K_ i\}\) which are in general position, where the branching index of each submanifold is \(d\). It is shown that these are rational homology manifolds. A formula for the eigenspace signatures of the cover in terms of the Hirzebruch \(L\)-class of \(M\) and 2-dimensional cohomology classes dual to the \(K_ i\) is given. Many singular algebraic varieties arise in this manner. As an application the signature of a \(d\)- fold branched cyclic cover of \(\mathbb{C} \text{P} (2k)\) along a collection of \(d\) hyperplanes in general position is calculated to be \(1+{d-1 \choose k+1}\). A formula for the stable tangent bundle of a branched cover along a single submanifold is also given.

57R20 Characteristic classes and numbers in differential topology
14F45 Topological properties in algebraic geometry
57M12 Low-dimensional topology of special (e.g., branched) coverings
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