A topological version of Bertini's theorem

Abstract

We give a topological version of a Bertini type theorem due to Abhyankar. A new definition of a branched covering is given. If the restriction πV : V → Y of the natural projection π : Y × Z → Y to a closed set V ⊂ Y × Z is a branched covering then, under certain assumptions, we can obtain generators of the fundamental group π1((Y × Z) \ V ). Introduction. In his book [1, pp. 349–356], Abhyankar proves an interesting theorem called by him a “Bertini theorem” or a “Lefschetz theorem”. The theorem expresses a topological fact in complex analytic geometry. The purpose of this paper is to restate this theorem and its proof in purely topological language. Our formulation reads as follows: Theorem 1. Let Z be a connected topological manifold (without boundary) modeled on a real normed space E of dimension at least 2 and let Y be a simply connected and locally simply connected topological space. Suppose that V is a closed subset of Y × Z and π : Y × Z → Y denotes the natural projection. Assume that πV = π|V : V → Y is a branched covering whose regular fibers are finite and whose singular set ∆ = ∆(πV ) does not disconnect Y at any of its points. Set X = (Y × Z) \ V and L = {p} × Z, where p ∈ Y \∆. If there exists a continuous mapping h : Y → Z whose graph is contained in X, then the inclusion i : L \ V ↪→ X induces an epimorphism i∗ : π1(L \ V )→ π1(X). We have adopted the following definition. For any topological spaces Y and Y ∗, a continuous, surjective mapping ψ : Y ∗ → Y is a (topological) branched covering if there exists a nowhere dense subset ∆ of Y such that ψ|Y ∗ \ ψ−1(∆) : Y ∗ \ ψ−1(∆)→ Y \∆ is a covering mapping. Notice that the singular set ∆ of the branched covering ψ is not unique, but there is a smallest ∆(ψ) among such sets. Clearly, ∆(ψ) is a closed subset of Y . Thus 1991 Mathematics Subject Classification: 57M12, 55Q52.